Fluid Flow:
Overview of Fluid Flow Analysis
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
Objectives
Module 1: Overview
Page 2

Become familiar with the underlying theory of fluid flow.

Understand fluid viscosity.

Differentiate between compressible and incompressible flow.

Examine the Navier-Stokes equation.

Understand how numerical methods apply.

Identify key design and simulation principles.

Learn from an example of Couette Flow and apply a what-if analysis.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
Introduction to Fluid Flow
Module 1: Overview
Page 3




“Fluid” is a generic term used to describe both liquids and gases.
Fundamental laws such as conservation of mass, momentum and
energy provide the equations that underlie these analyses.
In addition an Equation of State may also be used for finding
unknown variables such as density and temperature.
Complex equations mostly require numerical solutions.
Exact
Solutions
Diagram (not to scale or
proportion) approximating the
relative applicability of CFD
Numerical
Methods/
CFD
Experimental Techniques
/Regression Modelling
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Application of Computational Fluid Dynamics
(CFD)
Section 5 – Fluid Flow
Module 1: Overview
Page 4
The diversity of CFD has led to its extensive use in many applications:








Process and process equipment
Power generation, petroleum and environmental projects
Aerospace and turbomachinery
Automotive
Electronics / appliances /consumer products
HVAC / heat exchangers
Numerical
Methods/
Materials processing
CFD
Architectural design and fire research
Other
Methods
Today, CFD represents a major portion of fluid flow
solutions (dimensions/proportions approximate).
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
Underlying Theory
Module 1: Overview
Page 5
Energy equation
Conservative form of
Navier-Stokes equation
Continuity equation
Fluid Pressure and Velocity
are the two main variables of interest in fluid flow analysis.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
Understanding Viscosity
Module 1: Overview
Page 6






Viscosity is the measure of resistance to fluid flow.
Inviscid fluid is an ideal case in which viscous forces are absent.
Rarefied flow in the outer atmosphere can be approximated as a real
life example of inviscid flow.
Equations such as the Euler and Bernoulli equations ignore effects of
viscosity and thus are restricted to approximate analyses.
To analyze and predict flow behavior accurately, effects of viscosity
cannot be ignored.
Viscous Fluids can be classified into:
• Newtonian
• Bingham
© 2011 Autodesk
• Dilatant
• Plastic
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
Understanding Viscosity
Module 1: Overview
Page 7
 (N/m2)
Newtonian
(high Viscosity)
e.g. Honey
Bingham-plastic
e.g. Toothpaste
Pseudo-plastic
e.g. Styling Gel
Newtonian
(Low Viscosity)
e.g. Water
Dilatant
e.g. Putty
Strain rate (1/s)
Fluid viscosity varies in behavior from simple Newtonian fluids to more
complex Pseudo-plastic fluids.
 Common engineering fluids are Newtonian (e.g. water, steam, air, oils).

© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
Incompressible Flow
Module 1: Overview
Page 8




Incompressible flow is comparatively easy to solve.
As density is constant, fluid flow can be solved by continuity and
momentum equations alone.
For all practical cases, air flow with Mach number below 0.3 can be
treated as incompressible.
Similarly liquids, unless at extremely high pressure, can be treated as
incompressible.

Although no liquid is truly incompressible, it is a very accurate
approximation.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
Compressible Flow
Module 1: Overview
Page 9



For compressible flow, as density is variable, the energy equation
needs to be introduced, which relates density to temperature.
To solve for both these additional variables (density and
temperature), a separate equation is also required.
The Boussinesq approximation or Equation of State can be used to
relate density and temperature  , T 
   1   (T  T )

Where:
α is the coefficient of volume expansion.
ρo is the known value of density at temperature To
The study of sound waves in air and choked flow
in a converging diverging nozzle are common
examples of compressible flow.
The shock wave created by a supersonic jet aircraft is an example of compressible flow.
Image courtesy of US Air Force and Wikipedia.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
Types of Flow and Navier-Stokes Equation
Module 1: Overview
Page 10




Turbulent flow

Laminar flow
Compressible vs. Incompressible
Laminar vs. Turbulent
Steady vs. Unsteady
Navier-Stokes equations are the most generic equations able to apply
to the different kinds of flow as mentioned above (in 3D or 2D).


vs
e.g. blood flow, flow over aerofoil/hydrofoil, smoke/exhaust plume analysis
Navier-Stokes equations are fundamentally complex, but can take
different forms and be simplified depending upon the nature of flow.
Some exact solutions to the Navier-Stokes
Turbulent flow
equations exist for examples such as
Poiseuillie flow, Couette flow and Stokes
Aerofoil flow
boundary layer.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
The Navier-Stokes Equation
Module 1: Overview
Page 11
A short representation of the Navier-Stokes equation is its
vector form:
This form can be converted into an algebraic equation by replacing derivative terms
For incompressible flow:
.u  0
  density
u  velocity ( x  direction)
  divergence (vector  operator )
For the application of numerical methods, the above equation is discretized across a
domain that is broken up into small regions (discussed in detail in later section).
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
How Numerical Methods Apply: Part I
Module 1: Overview
Page 12

Expanding the Navier-Stokes equation:
  2u  2u  2u 
u
u
u
u
pˆ

ρ  ρu
 ρv  ρw
   μ


t
x
y
z
x
 x 2 y2 z 2 
 2v 2v 2v 
v
v
v
v
pˆ
ρ  ρu
 ρv  ρw
   μ 2  2  2 
t
x
y
z
y
y z 
 x
 2w 2w 2w 
w
w
w
w
pˆ
ρ
 ρu
 ρv
 ρw
   μ 2  2  2 
t
x
y
z
z
y
z 
 x
Local acceleration
Convective terms
Piezometric
pressure
gradient
Viscous term
The Cartesian form of the Navier-Stokes equation is given above. The spatial
derivates are replaced with approximate algebraic equivalents.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
How Numerical Methods Apply: Part II
Module 1: Overview
Page 13

The Navier -Stokes equation can be discretized into algebraic
u u
equations:
 u 
  
i 1
i 1
2  x
 x  i
ui  ui 1
 u 

 
x
 x  i
u u
 u 
   i 1 i
x
 x  i




Algebraic equations can be solved by several available indirect (or
iterative) numerical methods such as Gauss-Siedel or Jacobi iteration.
The Tridiagonal Matrix Algorithm (TDMA, or Thomas Algorithm) is a
direct method and an alternate to Gaussian Elimination to solve the
algebraic equations.
TDMA is easily programmable and a student can create code using
TDMA as the algorithm of choice for solving equations.
Further details for discretization are provided in the next module.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
Key Design and Simulation Principles
Module 1: Overview
Page 14


Convergence is analogous to a spiral,
where the locus of the solution moves
toward the center of the spiral and
hence successive computations arrive
closer to the exact answer.
Exact
Solution
The user has to stop the numerical
solution based upon a pre-determined
level of accuracy. Otherwise the
solution would continue iterating ever
closer toward the exact result without
reaching it.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
Performing Analysis
Module 1: Overview
Page 15
Convergence criteria:

Initial value


Multiplier / under-relaxation factor


Controls the speed of progress toward a solution.
Iterations


A good initial value for variables (speed and pressure) will result in fewer iterations.
The number of times the equations are processed.
Residual values

Indicator of differences of variables between two
successive iterations.
Residual

Iterations
A fair idea of the above mentioned terms can be grasped by solving simultaneous
algebraic equations through any iterative scheme (e.g., Gauss–Siedel, TDMA).
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
Example: Couette Flow (Steady State)
Module 1: Overview
Page 16

Couette Flow

Assumptions



Y
Model / geometric simplifications
Fluid properties (Constant vs variable parameters)
u0
Moving Plate
Boundary Conditions


Moving / stationary wall
Constant / variable pressure outlet / inlet
u
0
2
y
Stationary Plate
X
2
u
0
t
© 2011 Autodesk
Newtonian viscosity
Flow is steady
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
Exact solution to Couette
Flow is given by:
y
u ( y )  u0
h
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
What-If Analysis
Module 1: Overview
Page 17

The following parameters can be changed and flow behavior can be
investigated:
Upper plate velocity
 Viscosity
 Thickness between the plates



A video presentation for the steady flow module is available for
setting up Couette Flow in Autodesk Simulation Multiphysics
software.
By setting up the template for Couette flow as shown in the video,
multiple what-if scenarios can be investigated.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
Summary
Module 1: Overview
Page 18

This module covered the basics of fluid flow.

Fluid flow can be classified into compressible vs. incompressible,
steady vs. unsteady and laminar vs. turbulent.

This identification has to be made by the user before any analysis.

Fluid viscosity is a major factor among the flow parameters.

The Navier-Stokes equation is a general equation that can apply to
various kinds of fluid flow.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community
Section 5 – Fluid Flow
Summary
Module 1: Overview
Page 19

However, the Navier-Stokes equation consists of complex partial
differential equations, and thus numerical methods are applied for
practical solutions.

When numerical methods are applied, it is important to ensure that
the solution converges.

If the solution does converge, the user must self-determine where to
stop the calculation based on what accuracy is required.

Each successive computation brings the result closer to the actual
value, but never to an exact answer.
© 2011 Autodesk
Freely licensed for use by educational institutions. Reuse and changes require a note indicating
that content has been modified from the original, and must attribute source content to Autodesk.
www.autodesk.com/edcommunity
Education Community