# Computational Fluid Dynamics in Biological Systems

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```					Computational Fluid Dynamics
in Biomedical Systems

A Simple Introduction
Boyd Gatlin
What I Hope To Do Today
• Define some common terms
• Identify some applications of CFD in
biology and medicine
• Describe the steps and processes of CFD
• Outline a physically intuitive development
of sets of CFD equations
• Present a typical case in medical biology
Examples of Biomedical CFD
Applications
•   Heart pumping             •   Artificial organ design
•   Vessel graft evaluation   •   Vocal tract analysis
•   Blood flow                •   Microbe locomotion
•   Air flow in lungs         •   Perfusion in tissues
•   Joint lubrication         •   Life support systems
•   Cell-fluid interface      •   Nose and sinus flows
•   Tendon-sheath             •   Spinal fluid flow
•   Gas exchange              •   Cardiac valve design
What is a fluid?
• A fluid is any substance which deforms
continuously under a shearing stress
• Includes liquids and gasses
• Classified as Newtonian or non-Newtonian
– Water and most gasses are Newtonian
– Whole blood, synovia, mucus, ketchup,
toothpaste, etc., are not
What are some fluid properties?
• Mass density—if this remains constant in a
flow, the fluid is incompressible
• Viscosity—a measure of resistance to flow


– varies with temperature for Newtonian fluids
– varies with local flow conditions (strain and
shear) for non-Newtonian
What are some properties of the
flow itself?
• Laminar, transitional, or turbulent
• Compressible or incompressible
• Confined (contained in a conduit), partially
confined (open channel), or unconfined
(flow over an object in the open)
• Subsonic, supersonic, hypersonic
• Reacting or non-reacting
• Single-phase or multi-phase
What is CFD?
Computational Fluid Dynamics is a set of
procedures, carried out in sequence or in
parallel, by which the classical equations of fluid
motion, plus any auxiliary relations, are
approximated by large sets of algebraic
equations which are then solve numerically on
computers.
How does CFD differ from
‘numerical modeling’?
• CFD is simply the process of obtaining
numerical solutions to accepted classical
equations (Navier-Stokes, ca 1827)
• Numerical modeling usually involves the
development of equations of assumed form
and containing free constants derived from
experimental data
• CFD is the application of ‘first principles’ of fluid
physics
• ‘Modeling’ is required within CFD to:
– Compute fluid properties from empirical relationships
– Represent the effects of turbulence on the flow
– Include phenomena which cannot reasonably be
computed directly from first principles
– Account for chemical reactions and phase changes
• Both computational simulation and modeling
involve the numerical solution of very large sets of
equations on computers
CFD by the Numbers
I.     State the Problem
II.    Identify Physics & Chemistry
III.   Identify the Geometry Domain
IV.    Formulate the Mathematical Statement
V.     Discretize Equations & Geometry
VI.    Develop or Choose the Algorithm
CFD by the Numbers
VII. Develop or Select Software
VIII.Choose Computing Hardware
IX. Compute the Solution
X. Analyze & Interpret Results
A.   Storage and Manipulate Large Data Sets
B.   Post Processing and Visualization
Statement of Problem
• Clearly identified (What is the shearing
stress on an artery graft during systole?)
• Separated from irrelevant, extraneous or
non-essential issues
• Constrained so that key parameters and
variables can be isolated so much as
possible
Identification of Physics
• Physical principles related to the problem
• Statement of physical principles in
mathematical form
• Reduction of complexity to make amenable
to computation
• Identification of auxiliary relations
(turbulence model, non-Newtonian effects,
property variations, etc.)
Problem Geometry
• Set the physical boundaries; define the
domain
• Obtain data by measurement, digitized
scans, etc.
• Observe boundary conditions of physics
• Generate bounding surfaces (sets of cross
sections, algebraic models, etc.)
Mathematical Statement
• Generally, Navier-Stokes equations, or some
subset
• Equation of state (e.g., ideal gas law)
• Relationship between fluid and surfaces, and
between different fluids (mucus, porosity)
• Appropriate boundary and initial conditions
• Turbulence model
• Fluid property models and reaction equations
Discretization
• Divide the problem domain into a very large
number of very small regions using a grid
• Replace the classical differential equations
with sets of finite difference or finite
volume equations
• Balance accuracy needs with computing
capacity (3-D is very much bigger than 2-D)
A grid (mesh) is a system of uniquely identifiable
and indexed points in space upon who continuous
functions may be approximated through some
method of discretization.
• Structured         • Unstructured
• Orthogonal        • Non-orthogonal
• Uniform            • Non-uniform
• Boundary fitted  • Non-boundary fitted
• Rectilinear        • Non-rectilinear
• Fine              • Coarse
Typical CFD Grids

Structured   Unstructured
Grids Before Descartes
Discretization of Mathematics
• Finite difference approximation---replace derivatives with
ratios of differences at each grid point
– Mathematically intuitive
– Physics less obvious
• Finite volume formulation---apply physical ‘conservation
laws’ to each small volume formed by grid lines
– Mathematically simple, if not naïve
– Physically intuitive
• Finite element methods---a family of techniques that are
generally not physically intuitive
Discretization of Mathematics
• Finite difference approximation---replace derivatives with
ratios of differences at each grid point
– Mathematically intuitive
– Physics less obvious
• Finite volume formulation---apply physical ‘conservation
laws’ to each small volume formed by grid lines
– Mathematically simple, if not naïve
– Physically intuitive
• Finite element methods---a family of techniques that are
generally not physically intuitive
Simple Finite Difference Approximation
of a Derivative

u   u   u (i  1)  u (i )
;    ;
x   x   x(i  1)  x(i )
Finite Control Volume

Mass, energy,momentum    Mass,
inflow/outflow           energy,
momentum
stored

Net mass, energy and momentum entering the volume is
forced to balance that leaving and that stored in the FV
Algorithm
• Method by which the large system of
equations will be solved
• Choose based on mathematical character,
method of discretization; there is no ‘one
size fits all’ available
• Poor choices, based on available hardware
& software, may lead to failure
Software
• Software translates algorithm statements
into computer operations
• Ample CFD software available, much
unsuited for biomedical applications
• Poor choices may lead to complete failure
of the computational simulation
Computing Hardware
• PC vs workstation vs supercomputer
• Serial vs parallel platform
• Considerations
– Physical size of problem (number of grid
points)
– Time scales involved
– Availability
Solution
• Goal is to obtain solution to the problem not
just to the mathematical statement
• Expect false starts, adjustment of boundary
& initial conditions
• Monitor intermediate results for
reasonableness
• Validate solution method on problem for
which solution is known
Analysis & Interpretation
• Post process solution data
– Calculate derivative variables (pressure
– Compare with data or similar solutions
• Use visualization tools
– Examine distributions, look for trends
– Animate in time or space
Respiratory Flow in A
Bifurcation: A Case Study
Problem Statement
Flow development in airways of the lung is not understood
well enough by medical science and cannot be satisfactorily
measured.
Mathematical Model
• Air flow at low speeds is well predicted by
the incompressible form of the Navier-
Stokes Equations
• Air at low speeds is a constant-property,
Newtonian fluid
• Flow is mostly laminar in small airways
• Rigid walls are a valid first approximation
Geometric Domain
• Airway tree is hopelessly complex
• Branching pattern is not reproducible
Surfaces of Selected Model
Discretization of Geometry

Structured, finite-volume grid
Discretizaton & Algorithm
• Finite volume formulation (enforce
conservation laws)
• Implicit time stepping (solution is computed
at new time level)
– Inherently stable because closer to physics
– Larger time steps possible
• Newton iteration to converge each time step
Visualization of Results
Animation of Simulation of Flow
in an Asymmetric Bifurcation
• Lung airways do not
branch symmetrically
• Effects of axial
curvature important
• Real flow does not
enter smoothly
• Inlet & outlet
conditions unknown
Oscillatory Flow in a Lung-Like Bifurcation

• Colors represent fluid
speed
• Skewness due to
friction and curvature
• Details not available
from experiment

Colored profiles act like porous membranes which stretch in proportion to the local fluid speed.

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