Purpose of Modeling
Before we embark on our journey to develop mathematical models of fluid motion, we
should pause and ask some questions.
Question: Why do we need mathematical models?
Answer: In science and engineering, mathematical modeling gives us a way of describing physical processes, so we
gain insight into those processes. For example, why does the pressure go down as fluid flows in a pipe? What causes
lift on an aircraft wing? How much force will water exert on a wall? Models of fluid motion can answer these
questions and many more.
Question: What are some requirements of these models?
Answer: The principal requirement is that the models are reasonably accurate (though possibly simplified)
representations of the physical processes. In addition, these models should be solvable by analytical or numerical
means. This will impose some requirements on the mathematical model itself, specifically that the mathematical
problems are “well-posed.”
2
Simplified Models
• Models need not represent all physical processes seen in the real world. Simplified models can often yield very
useful results and represent limiting cases for observed fluid behavior.
• Some important simplifications in fluid dynamics include:
- Assuming steady flow conditions Æ Ignore small scale unsteady fluctuations
- Simplifying turbulent flow Æ Modeling scales of turbulence to render steady-state models.
- Assuming viscous effects are confined to boundaries Æ Boundary layer model
- Assuming the flow is incompressible Æ Ignore compressibility effects
- Ignoring viscous effects entirely Æ Fluid’s momentum is not impacted significantly by viscous regions
- Treating 3D flow as 2D or 1D Æ Compressible nozzle flows can be approximated as 1D
- Assuming fluid properties (e. g., viscosity, density, specific heat) are constant Æ Valid for low-speed flows without any
significant heat transfer
• One important reason to make simplifying assumptions is to make our mathematical model tractable! There is
no need to include all physical effects if only a certain aspect of the fluid’s motion is of interest.
- For example, we can use inviscid flow models to predict the lift for an aircraft wing with reasonable accuracy.
3
Differential Equations
• Mathematical modeling of physical phenomena typically can be cast as one or more differential
equations.
• These equations will represent some aspect of the physics, for example conservation of mass,
momentum, energy, species, etc., and arise from fundamental laws such as Newton’s Second Law of
Motion.
• We need to have as many equations as unknowns. For isothermal (constant temperature),
incompressible (constant density) flows, the primary variables will be the three velocity components
and pressure. So for 3D flow models, we will need four equations. If thermal and compressibility
effects are important, then additional equations will be introduced (the conservation of energy and
an equation of state for the fluid).
• In addition, we will need to prescribe boundary conditions for all surfaces of our system. These
boundary conditions will contain the essential inputs to our model (e.g., boundary flow rates,
pressures, velocities, etc.) and, more importantly, provide a unique solution to our model. Without
uniqueness, our model is useless!
4
Modeling and Simulation
• The governing differential equations of fluid dynamics form the
basis for modern computer modeling, known generically as
computational fluid dynamics, or CFD.
• Until the 1960s and 1970s, solutions to the equations of fluid
dynamics were typically based on approximations and
simplifications that made solutions tractable for analytical
methods for differential equations. When powerful
computational resources became available, numerical solution
techniques were devised and general purpose CFD codes Volume
developed to provide general solutions for practical problems. Fraction
[Hg]
Today, CFD simulation (and tremendous computer power) is
pervasive. As a result, CFD has now become embedded as part
of the analysis workflow for engineers and scientists. Part of the
purpose of this course is to expose you to this now common
form of fluid dynamics analysis using the CFD software
Ansys Fluent.
5
Model Validation
• Is it enough to develop a model of fluid motion and solve it? No! We must always compare the
solution obtained from the model to the real world, a process called validation.
• For example, we may solve a mathematical model for the pressure drop due to viscous flow in a pipe
and find that the result deviates 10% from physical measurements. Is this close enough? That
depends entirely on the application — in some cases yes, in others no.
• However, validation also provides guidance on how to improve the model, by including additional
physics and removing simplifying assumptions.
• We should also not neglect the fact that our boundary conditions are inputs to the model, and as such
their accuracy will directly affect the accuracy of a model solution.
• It is therefore important to observe the symbiotic relationship between experimental fluid dynamics
(where data is collected from carefully performed experiments) and analytical fluid dynamics, which
relies on mathematical modeling.
6
Summary
• We will now embark on our study of the governing equations of fluid dynamics.
• For this course, we will only consider some of the more common, relatively simplified forms of the
equations, with some comments on the more complex forms.
• However, the modeling process will remain the same, regardless of the complexity of the physical
processes. Namely:
o Define your problem precisely.
o Make reasonable simplifying assumptions.
o Develop equations and associated boundary conditions for the model.
o Solve the model with appropriate mathematical techniques (either exactly or numerically).
o Analyze the results to see what insights the solution provide to the physics.
