An Introduction To

Computational Fluid Dynamics (CFD)

• 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 Turbo-machinery is shown below
• First step of CFD is to Identify the flow physics involved in pump:turbo-machinery, turbulent flow, multi-phase 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), multi-phase flow (particle and flue gas), reactive flow (combustion)

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•  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.
  • Viscous Models and Fluid Dynamics
  • Turbulent Flow Modeling
  • Turbulent Heat Transfer Modeling
  • Types of Multiphase and Its CFD Modeling
  • Turbulent Combustion Modeling
  • CFD Modeling of Turbo-machinery

 Laminar and Turbulent flow

• In general, the Reynolds number is used to determine the flow whether it is laminar or turbulent

• Example 3:  Water channel flows show that laminar flows in the channel and turbulent around the gate due to increase in velocity

• 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

• Condensation consists of two phases water and steam

• Non-reactive 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.

• Solid combustion comprises the pyrolysis stage and devolatilization, reaction with oxygen

• 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.

• 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

 General form of governing equation 

The general form of governing equation (in compact form)

•  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).

Fig.13 Computational domain defined for a circular pipe

 Steps for CFD Modelings

• Pre-processing comprises making geometry and meshing
• Solver selection of physical models,  governing equations, solution procedures
• Post-processing of CFD results
• Steps for CFD modelings are given as below:
  1. Define the objectives of modeling
  2. Create the geometry  or computational domain using the CAD tools like Space claim, solid works
  3. Create the meshing using the meshing software like ANSYS Meshing, Hyper mesh
  4. Set up the solver and physical models
  5.  Compute and monitor the solution
  6. Examine and save the results in plots, contours, and flux reports
  7.  Consider revisions to the numerical or physical model parameters, if necessary

CFD Software

• 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  well-practiced 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

Semi-Comprehensive 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, 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 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

Structured vs non-structured 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 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

• • High-quality 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 non-zero 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 poor quality mesh, the solutions can be obtained by controlling the relaxation factor or selecting near-wall gradient schemes
  • Try to keep at least three-element in narrow gaps which is a shadow or wake region.
  • Add more elements (at least 10 cells) in high-velocity fluid zones to better numerical convergence

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 time-dependent. 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  the use of curvilinear transforms the method can be extended to domains that can not be easily represented using  brick-shaped 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’s 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 non-uniform 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 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 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 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, Open FOAM)

•  For ANSYS FLUENT  click here: FLUENT_NUMERICAL_ METHODS
• For OpenFOAM, click here: OpenFOAM_Numerical_Scheme

Finite  Element Methods (FEM)

• This numerical method 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 double-edged 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?

• Flow direction-dependent Øe
• This approximation is of first-order (O(Δx))
•   Numerically diffusive

 Unsteady Term discretization

2) Implicit or backward Euler method:: accuracy in order O(Δt)

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
  • Trid-iagonal matrix algorithm (TDMA)

2) Indirect or iterative methods:

• • Jacobi method
  • Gauss-Seidel method
  • Successive over-relaxation (SOR)
  • Conjugate gradient method (CG)
  • Multi-grid 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 Lagrangian-Eulerian (ALE).

SIMPLE Algorithm

• Semi-implicit method for Pressure-linked Equations

1. 1. 1.Advance to the next iteration t(n)=t(n+1)
   2. Initialize u(n+1) and p(n+1) using latest available values of u and p
   3. Construct the momentum equations
   4. Under-relax the momentum matrix
   5. Solve the momentum equations to obtain a prediction for u (n+1)
   6. Construct the Pressure equation
   7. Solve the pressure equation for p(n+1)
   8. Under-relax p(n+1)
   9. Correct the velocity for u(n+1)
   10. If not converged, go back to step 2
   11. Correct the fluxes for ϕ(n+1)

b) SIMPLEC:Semi-Implicit Method for Pressure Linked Equations-Consistent

• The pressure-correction under-relaxation 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)

d) Fractional step method

• • • This method solves  the incompressible Navier-Stokes 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: 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

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), Multi-phase (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: no-slip (fluid is at rest at the wall) or free-slip (no frictional losses at the boundary).
• Wall can be slip or no-slip type
• Wall can be stationary or moving wall (like rotor for turbo-machinery)
• Input to be specified at the wall: Momentum (velocity, roughness), Thermal (temperature, convective HTC, emissivity and shell conduction), Species, Multi-phase

5) Periodic (cyclic) condition:

• 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