Fundamentals of CFD: Navier-Stokes Equations, Unimore Presentation

GRUMO Internal Combustion Engine Research Group

Sustainable Industrial Engineering

Theory and Simulation of Industrial Fluid Machines A.A. 2024/2025

Fundamentals of CFD

Lecturer: Fabio Berni (fabio.berni@unimore.it)

References for CFD

V Teacher's notes Software manual Versteeg H.K., Malalasekera W., "An Introduction to Computational Fluid Dynamics" (Longman, 1995) V Ferziger J. H., Peric M., "Computational Methods For Fluid Dynamics" (Springer, 2002)

Software Tools for CFD

V 3D-CFD simulation: V SIMCENTER STAR-CCM+ (SIEMENS DISW) CFD model files Available during the course on:

• Teams
• USB pendrives

What is Computational Fluid Dynamics (CFD)?

Computational fluid dynamics (CFD) is the science of predicting fluid flow, heat transfer, mass transfer, chemical reactions, and related phenomena by solving the mathematical equations which govern these processes using a numerical procedure. The result of CFD analyses is relevant engineering data used in: - Conceptual studies of new designs. - Detailed product development. - Troubleshooting & redesign. CFD analysis complements testing and experimentation: - Reduces the total effort required in the laboratory. Applications of CFD are numerous: - Flow and heat transfer in industrial processes (boilers, heat exchangers, combustion equipment, pumps, blowers, piping, etc.). Ventilation, heating, and cooling flows in buildings. - Aerodynamics of ground vehicles, aircraft, missiles. - Film coating, thermoforming in material processing applications. - Flow and heat transfer in propulsion and power generation systems. - Heat transfer for electronics packaging applications. - Fuel cells and electro-chemical devices (e.g. electrolyzers) - And many more!

Advantages of CFD

1. Relatively low cost. - Using physical experiments and tests to get essential engineering data for design can be expensive. - CFD simulations are relatively inexpensive, and costs are likely to decrease as computers become more powerful.
2. Speed. - CFD simulations can be executed in a short period of time. - Engineering data can be introduced early in the design process.
3. Ability to simulate real conditions. - Many flow and heat transfer processes can not be (easily) tested, e.g. hypersonic flow. - CFD provides the ability to theoretically simulate any physical condition. Ability to simulate ideal conditions. - CFD allows great control over the physical process and provides the ability to isolate specific phenomena for study. Example: a heat transfer process can be idealized with adiabatic, constant heat flux, or constant temperature boundaries.
4. Comprehensive information. - Experiments only permit data to be extracted at a limited number of locations in the system (e.g. pressure and temperature probes, heat flux gauges, LDV, etc.). - CFD allows the analyst to examine a large number of locations in the region of interest and yields a comprehensive set of flow parameters for examination.

Limitations of CFD

1. Physical models. - CFD solutions rely upon physical models of real world processes (e.g. turbulence, compressibility, chemistry, multiphase flow, etc.): CFD can only be as accurate as the physical models on which it is based.
2. Numerical errors. Solving equations on a computer invariably introduces numerical errors: - Round-off error: due to finite word size available on the computer. Round-off errors will always exist (though they can be small in most cases). - Truncation error: due to approximations in the numerical models. Truncation errors will decrease as the grid is refined. Mesh refinement is one way to deal with truncation error.
3. Boundary conditions. - The accuracy of a CFD solution is only as good as the boundary conditions provided to the numerical model. Example: flow in a duct with sudden expansion. If flow is supplied to domain by a pipe, you should use a fully-developed profile for velocity rather than assume uniform conditions. Computational Domain Computational Domain Uniform Inlet Profile Fully Developed Inlet Profile poor better

CFD Workflow

Analysis begins with a mathematical model of a physical problem. · Conservation of matter, momentum, and energy must be satisfied throughout the region of interest. · Fluid properties are modeled empirically. · Simplifying assumptions are made in order to make the problem tractable (e.g., steady-state, incompressible, inviscid, two-dimensional). · Appropriate initial and boundary conditions for the problem are needed and imposed. CFD applies numerical methods (called discretization) to develop approximations of the governing equations of fluid mechanics in the fluid region of interest. Domain is discretized into a finite set of control volumes or cells : the collection of cells is called the grid. - General conservation (transport) differential equations for mass, momentum, energy, etc., are discretized into algebraic equations. - The set of algebraic equations are solved numerically (on a computer) for the flow field variables at each node or cell. - System of equations is solved simultaneously to provide solution. - The solution is post-processed to extract quantities of interest (e.g. lift, drag, torque, heat transfer, separation, pressure loss, etc.). Filling Nozzle Bottle Domain for bottle filling problem. Mesh for bottle filling problem.

Design and Create the Grid

1. Should you use a quad/hex grid, a tri/tet grid, a hybrid grid, or a non-conformal grid?
2. What degree of grid resolution is required in each region of the domain?
3. How many cells are required for the problem?
4. Will you use adaption to add resolution?
5. Do you have sufficient computer memory?

Tri/tet vs. Quad/hex Meshes

tetrahedron pyramid triangle 1 hexahedron prism or wedge quadrilateral · For simple geometries, quad/hex meshes can provide high- quality solutions with fewer cells than a comparable tri/tet mesh. · For complex geometries, quad/hex meshes show no numerical advantage, and you can save meshing effort by using a tri/tet mesh. · Hybrid meshes are a good compromise: specific regions can be meshed with different cell types. Both efficiency and accuracy are enhanced relative to a hexahedral or tetrahedral mesh alone. arbitrary polyhedron

Set Up the Numerical Model

For a given problem, you will need to: - Select appropriate physical models: turbulence, combustion, multiphase, etc. - Define material properties: Fluid, Solid, Mixture. - Prescribe operating conditions: boundary conditions at all boundary zones and an initial solution. - Set up solver controls and convergence monitors.

Compute the Solution

1. The discretized conservation equations are solved iteratively. A number of iterations are usually required to reach a converged solution. 2. Convergence is reached when: - Changes in solution variables from one iteration to another are negligible. - Residuals provide a mechanism to help monitor this trend. - Overall property conservation is achieved. 3. The accuracy of a converged solution is dependent upon: - Appropriateness and accuracy of the physical models, grid resolution and independence - Problem setup.

Consider Revisions to the Model

· Are physical models appropriate? - Is flow turbulent? - Is flow unsteady? - Are there compressibility effects? - Are there 3D effects? - Are boundary conditions correct? · Is the computational domain large enough? - Are boundary conditions appropriate? - Are boundary values reasonable? · Is grid adequate? - Can grid be adapted to improve results? - Does solution change significantly with adaption, or is the solution grid independent? - Does boundary resolution need to be improved?

Governing Equations Describing Flow Motion: Navier-Stokes Equations

They are based on 3 fundamentals of fluid physics: mass conservation (also known as continuity) , momentum conservation (also known as Newton's second law) and energy conservation (a.k.a. first law of thermodynamics). IMPORTANT: the fluid is considered as a "continuum", since the analyses are carried out on "macroscopic" scales (i.e. bigger than 1um) and the molecular structure of the fluid and the molecular motion can be ignored and disregarded. Fluid behavior is described in terms of macroscopic properties such as velocity, pressure, density, temperature and their spatial and temporal derivatives, all considered as average values over a huge number of single molecules which can be approximated as a point in space (or a single fluid particle), defined as the smallest fluid element which is not influenced by the behavior of the single molecules. The "continuum" formulation is valid when the Knudsen number, defined as Kn = = (where I is the molecular mean free path and L is the characteristic length of the body), is lower than 0.01 For high Reynolds number, the Knudsen number can be written as Kn = " w Re 2 where M is the Mach number, y is the ratio of specific heats, and the Reynolds number is dependent on L.