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
•   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
    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
• ‘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
      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
• 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
• 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
• 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,

Net mass, energy and momentum entering the volume is
forced to balance that leaving and that stored in the FV
• 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 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
  – Whether problem is steady or unsteady
  – Time scales involved
  – Availability
• 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
• Validate solution method on problem for
  which solution is known
     Analysis & Interpretation
• Post process solution data
  – Calculate derivative variables (pressure
    gradients, shear stresses, etc.)
  – 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
        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
• 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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