Chapter 3

Introduction to Computational Fluid Dynamics

Fluid (gas and liquid) flows are governed by partial differential equations which represent conservation laws for the mass, momentum, and energy. Computational Fluid Dynamics (CFD) is the art of replacing such PDE systems by a set of algebraic equations which can be solved using digital computers.

Computational Fluid Dynamics (CFD) provides a qualitative (and sometimes even quantitative) prediction of fluid flows by means of

? Mathematical modeling (partial differential equations)

? Numerical methods (discretization and solution techniques)

? Software tools (solvers, pre- and post-processing utilities)

CFD enables scientists and engineers to perform ‘numerical experiments’ (i.e. computer simulations) in a ‘virtual flow laboratory’

Components

CFD uses a computer to solve the mathematical equations for the problem at hand. The main components of a CFD design cycle are as follows:

? The human being (analyst) who states the problem to be solved

? Scientific knowledge (models, methods) expressed mathematically

? The computer code (software) which embodies this knowledge and provides detailed instructions (algorithms) for

? The computer hardware which performs the actual calculations

? The human being who inspects and interprets the simulation results

CFD is a highly interdisciplinary research area which lies at the interface of physics, applied mathematics, and computer science.

CFD Analysis Process

1. Problem statement

2. Mathematical model

3. Mesh generation

4. Space discretization

5. Time discretization

6. Iterative solver

7. CFD software

8. Simulation

9. Post processing

10. Verification

Mathematical Model

1. Choose a suitable flow model (viewpoint) and reference frame.

2. Identify the forces which cause and influence the fluid motion.

3. Define the computational domain in which to solve the problem.

4. Formulate conservation laws for the mass, momentum, and energy.

5. Simplify the governing equations to reduce the computational effort:

? use available information about the prevailing flow regime

? check for symmetries and predominant flow directions (1D/2D)

? neglect the terms which have little or no influence on the results

? model the effect of small-scale fluctuations that cannot be captured

? incorporate a prior knowledge (measurement data, CFD results)

? Add constitutive relations and specify initial/boundary conditions.

Discretization Process

The PDE system is transformed into a set of algebraic equations

? Mesh generation (decomposition into cells/elements) structured or unstructured, triangular or quadrilateral?

? CAD tools + grid generators (Delaunay, advancing front)

? mesh size, adaptive refinement in ‘interesting’ flow regions

2. Space discretization (approximation of spatial derivatives)

? finite differences/volumes/elements

? high- vs. low-order approximations

3. Time discretization (approximation of temporal derivatives)

? explicit vs. implicit schemes, stability constraints

? local time-stepping, adaptive time step control

Turbulence Models

The standard k-? model

The standard k-? model is a semi-empirical model based on model transport equations for the turbulence kinetic energy (k) and its dissipation rate (?). The model transport equation for k is derived from the exact equation, while the model transport equation for ? was obtained using physical reasoning and bears little resemblance to its mathematically exact counterpart.

These default values have been determined from experiments with air and water for fundamental turbulent shear flows including homogeneous shear flows and decaying isotropic grid turbulence. They have been found to work fairly well for a wide range of wall-bounded and free shear flows.

CFD Simulations

The computing times for a flow simulation depend on

? the choice of numerical algorithms and data structures

? linear algebra tools, stopping criteria for iterative solvers

? discretization parameters (mesh quality, mesh size, time step)

? cost per time step and convergence rates for outer iterations

? programming language (most CFD codes are written in Fortran)

? Many other things (hardware, vectorization, parallelization etc.)

? The quality of simulation results depends on

? the mathematical model and underlying assumptions

? approximation type, stability of the numerical scheme

? mesh, time step, error indicators, stopping criteria

Post-processing and Analysis

Post processing of the simulation results is performed in order to extract the desired information from the computed flow field

? calculation of derived quantities (stream function, vortices)

? calculation of integral parameters (lift, drag, total mass)

? visualization (representation of numbers as images)

? 1D data: function values connected by straight lines

? 2D data: streamlines, contour levels, color diagrams

? 3D data: cutline, cut planes, iso surfaces, iso volumes

? Arrow plots, particle tracing, animations.

? Systematic data analysis by means of statistical tools

? Debugging, verification, and validation of the CFD model

Advantages

? Design and analysis, and optimization of body shapes.

? A few simulations at one design can reveal merits of one variant over the other.

? Quicker and less expensive changes to configuration and their numerical simulation.

? Smoothing of the configuration to reduce pressure drag levels.

? Detailed flow field information.

SOFTWARE USED

? Created geometry and domain in ANSYS workbench.

? Numerical Investigation of the flatplate model was carried out using solver FLUENT module of ANSYS.

PROCEDURE FOR FLOW ANALYSIS

1. Creating geometry in ANSYS workbench.

2. Create domain.

3. Add material and define boundaries.

4. Mesh the region between domain and geometry (structured Meshing).

5. Refine the mesh near the flat plate, at the top of the surface and at the bottom to ensure appropriate flow capturing.

6. Check the mesh quality and smooth mesh if required.

7. Define boundary conditions, initialize the flow and set up solver parameters using FLUENT-Pre.

9. Run Solver.

10. Calculate pressure, velocity and drag variations using FLUENT – Post.

Flat plate

MESHING DETAILS

Inflation layer over the flat plate

BOUNDARY CONDITIONS

Velocity Inlet Boundary Condition

Velocity inlet boundary conditions are used to define the flow velocity, along with all relevant scalar properties of the flow, at flow inlets.

The values of velocity inlet boundary conditions in FLUENT are given in the table below.

Symmetry Boundary Condition

Symmetry boundary conditions are used when the physical geometry of interest, and the expected pattern of the flow/thermal solution, has mirror symmetry. While analyzing an isolated flat plate model, since the model is symmetrical along its length only half model is used for the CFD analysis. The center plane which divides the model into two half’s, is assigned symmetry boundary condition, since the flow would be symmetrical on both sides under steady state conditions. This saves considerable amount of time for meshing and running the CFD simulations.

Wall Boundary Condition

The flat plate are assigned the no-slip wall boundary conditions. These conditions are used to bound fluid and solid regions. In viscous flows, the no-slip boundary condition is enforced at walls by default. The same is used for the current study on external aerodynamics. No modified effects are used i.e. default values are used and very fine meshes around the model are used to record the viscous effects.

Material

The medium flowing through the system is air. The properties of air for this analysis are taken at an ambient temperature having a viscosity of 1.7894*105 and density of 1.225 kg/m3

Solution Methods

Standard k-epsilon model with non-equilibrium wall function for near wall treatment is used. Solution method utilizes pressure-velocity coupling scheme as coupled with gradient: least square cell based method, pressure as linear and setting the momentum, turbulence kinetic energy, turbulence dissipation rate as second order upwind for the 1000 iterations. Moreover, in solution controls momentum is taken as 0.7, pressure as 0.3 and turbulence viscosity factor as 0.8 for all 10000 iterations.

M=8 1deg Static Pressure vs position M= 1.2 1 deg static pressure vs position

M= 0.8 1 deg static pressure vs position

M= 8 0 deg static pressure vs position M= 1.2 0 deg static pressure vs position

M= 0.8 0 deg static pressure vs position

M= 8 3deg static pressure vs position M= 1.2 3 deg static pressure vs position

M= 0.8 3 deg static pressure vs position

M= 8 5 deg static pressure vs position M= 1.2 5 deg static pressure vs position

M= 0.8 5deg static pressure vs position

M= 8 0deg static pressure contour M= 1.2 0deg static pressure contour

M= 0.8 0deg static pressure contour

M= 8 0deg Relative x velocity contour M= 1.2 0deg Relative x velocity contour

M= 0.8 0deg Relative x velocity contour

M= 8 1 deg static pressure contour M= 1.2 1deg static pressure contour

M= 0.8 1deg static pressure contour

M= 8 1deg Relative x velocity contour M= 1.2 1deg Relative x velocity contour

M= 0.8 1deg Relative x velocity contour

M = 8 3deg static pressure contour M = 1.2 3deg static pressure contour

M= 0.8 3deg static pressure contour

M= 8 3deg Relative x velocity contour M= 1.2 3deg Relative x velocity contour

M= 0.8 3deg Relative x velocity contour