CFD in Engineering Applications

Ashoke De
Associate Professor, Dept. Of Aerospace Engineering, IITK
12-3-2019

INDIAN INSTITUTE OF TECHNOLOGY KANPUR Computational Propulsion Laboratory

  CAE Background

 Basics

 CFD – what, where and why ?

 CFD - fundamentals

 CFD – essentials

 Examples
Ashoke De (2) Computational Propulsion Laboratory, IITK
 Introduction

What is Computational Fluid Dynamics(CFD)?

 Why and where use CFD?
 Physics of Fluid
 Navier-Stokes Equation
 Numerical Discretization
 Grids
 Boundary Conditions
 Numerical Staff

Ashoke De (3) Computational Propulsion Laboratory, IITK

 Role of CAE (CFD)
 Computer-Aided Engineering (CAE) is the broad usage
of computer software to aid in engineering analysis tasks.

 Finite Element Analysis (FEA)

 Computational Fluid Dynamics (CFD)
 Multi-body dynamics (MBD)
 Optimization

 Software tools that have been developed to support these

activities are considered CAE tools.

 The term encompasses simulation, validation, and optimization

of products and manufacturing tools.

In the future, CAE systems will be major providers of information

to help support design teams in decision making !!

Ashoke De (4) Computational Propulsion Laboratory, IITK

 What is CFD?
 Computational fluid dynamics (CFD) deals with solution of fluid
dynamics and heat transfer problems using numerical techniques.

 CFD is an alternative to measurements for solving large-scale fluid

dynamical systems.

 CFD has evolved as a design tool for various industries namely

Aerospace, Mechanical, Auto-mobile, Chemical, Metallurgical,
Electronics, and even Food processing industries.

 CFD is becoming a key-element for computer-aided designs in

industries across world over.

Ashoke De (5) Computational Propulsion Laboratory, IITK

 What is CFD?

Fluid
Fluid Mechanics Problem Comparison &
Analysis
Physics of Fluid Simulation Results
C
Mathematics F Computer

Navier-Stokes Equations D
Computer Program
Numerical Programming
Methods Geometry Language
Discretized Form Grids

Ashoke De (6) Computational Propulsion Laboratory, IITK

 Phase of CFD
 Pre-processing – defining the geometry model, the physical model and
the boundary conditions
 Computing (usually performed on high powered computers (HPC))
 Post-processing of results (using scientific visualization tools &
techniques )

Iterative process !!
Ashoke De (7) Computational Propulsion Laboratory, IITK
 CFD
 CFD is the “science” of predicting fluid behaviour
• Flow field, heat transfer, mass transfer, chemical
reactions, etc…
• By solving the governing equations of fluid flow using a numerical
approach (computer based simulation)

 The results of CFD analyses

• Represent valid engineering data that may be used for
• Conceptual studies of new designs (with reduction of lead time and
costs)
• Studies where controlled experiments are difficult to perform
• Studies with hazardous operating conditions
• Redesign engineering

 CFD analyses represent a valid

• Complement to experimental tests
• Reducing the total effort required in laboratory tests

Ashoke De (8) Computational Propulsion Laboratory, IITK

 Introduction

 What is Computational Fluid Dynamics(CFD)?

 Why and where use CFD?
 Physics of Fluid
 Navier-Stokes Equation
 Numerical Discretization
 Grids
 Boundary Conditions
 Numerical Staff

Ashoke De (9) Computational Propulsion Laboratory, IITK

 Why use CFD?
• Analysis and Design
Simulation-based design instead of “build & test”
More cost effectively and more rapidly than with experiments
CFD solution provides high-fidelity database for interrogation of flow field
Simulation of physical fluid phenomena that are difficult to be
measured by experiments
Scale simulations (e.g., full-scale ships, airplanes)
Hazards (e.g., explosions, radiation, pollution)
Physics (e.g., weather prediction, planetary boundary layer, stellar
evolution)

• Knowledge and exploration of flow physics

Ashoke De (10) Computational Propulsion Laboratory, IITK

 Why use CFD?

Simulation(CFD) Experiment

Cost Cheap Expensive

Time Short Long

Scale Any Small/Middle

Information All Measured Points

Repeatable All Some

Security Safe Some Dangerous

Ashoke De (11) Computational Propulsion Laboratory, IITK

 Why use CFD?
 Computers built in the 1950s performed limited floating point
operations per second, i.e. only few hundred arithmetic operations
per second.
 Computers that are manufactured today have teraflops rating
where tera is a trillion and flops is an abbreviation for floating
point operations per second.
 While computer speed has increased at a tremendous rate,
computer cost has fallen significantly.
 It is revealed that the computational cost has been reduced by
approximately a factor of 10 every 8 years.
 Today a desktop machine can do the job of “mainframe” machines
of 1980s.

Ashoke De (12) Computational Propulsion Laboratory, IITK

 Where is CFD used? (Aerospace)

• Where is CFD used?

– Aerospace
– Appliances
– Automotive
F18 Store Separation
– Biomedical
– Chemical Processing
– HVAC&R
– Hydraulics
– Marine
– Oil & Gas
– Power Generation
– Sports Wing-Body Interaction Hypersonic Launch
Vehicle
Source: internet
Ashoke De (13) Computational Propulsion Laboratory, IITK
 Where is CFD used? (Appliances)

• Where is CFD used?

– Aerospace
– Appliances
– Automotive
– Biomedical
– Chemical Processing
– HVAC&R
– Hydraulics
– Marine
– Oil & Gas Surface-heat-flux plots of the No-Frost
refrigerator and freezer compartments helped
– Power Generation BOSCH-SIEMENS engineers to optimize the
– Sports location of air inlets.

Source: internet
Ashoke De (14) Computational Propulsion Laboratory, IITK
 Where is CFD used? (Automotive)

• Where is CFD used?

– Aerospace
– Appliances
– Automotive
– Biomedical
– Chemical Processing External Aerodynamics Undercarriage
Aerodynamics
– HVAC&R
– Hydraulics
– Marine
– Oil & Gas
– Power Generation
– Sports
Interior Ventilation
Engine Cooling
Source: internet
Ashoke De (15) Computational Propulsion Laboratory, IITK
 Where is CFD used? (Biomedical)

• Where is CFD used?

– Aerospace
– Appliances
– Automotive
– Biomedical
Medtronic Blood Pump
– Chemical Processing
– HVAC&R
– Hydraulics
– Marine
– Oil & Gas
Temperature and natural
– Power Generation convection currents in the eye
following laser heating.
– Sports
Spinal Catheter
Source: internet
Ashoke De (16) Computational Propulsion Laboratory, IITK
 Where is CFD used? (Chemical Processing)

• Where is CFD used?

– Aerospace
– Appliances
– Automotive Polymerization reactor vessel - prediction
– Biomedical of flow separation and residence time
effects.

– Chemical Processing
– HVAC&R
– Hydraulics
– Marine
– Oil & Gas
Twin-screw extruder
– Power Generation modeling

– Sports Shear rate distribution in twin-

screw extruder simulation

Source: internet
Ashoke De (17) Computational Propulsion Laboratory, IITK
 Where is CFD used? (HVAC&R)

• Where is CFD used?

– Aerospace
– Appliances
– Automotive
Particle traces of copier VOC emissions
– Biomedical colored by concentration level fall
Streamlines for workstation behind the copier and then circulate
– Chemical Processing ventilation through the room before exiting the
exhaust.
– HVAC&R
– Hydraulics
– Marine
– Oil & Gas
– Power Generation
– Sports Flow pathlines colored by
pressure quantify head loss
Mean age of air contours indicate
in ductwork
location of fresh supply air Source: internet
Ashoke De (18) Computational Propulsion Laboratory, IITK
 Where is CFD used? (Hydraulics)

• Where is CFD used?

– Aerospace
– Appliances
– Automotive
– Biomedical
– Chemical Processing
– HVAC&R
– Hydraulics
– Marine
– Oil & Gas
– Power Generation
– Sports
Source: internet
Ashoke De (19) Computational Propulsion Laboratory, IITK
 Where is CFD used? (Marine)

• Where is CFD used?

– Aerospace
– Appliances
– Automotive
– Biomedical
– Chemical Processing
– HVAC&R
– Hydraulics
– Marine
– Oil & Gas
– Power Generation
– Sports
Source: internet
Ashoke De (20) Computational Propulsion Laboratory, IITK
 Where is CFD used? (Oil & Gas)

• Where is CFD used?

– Aerospace
Volume fraction of gas
– Appliances
– Automotive
– Biomedical
Flow vectors and pressure Volume fraction of oil
– Chemical Processing distribution on an offshore oil rig
– HVAC&R
– Hydraulics
– Marine Volume fraction of water

– Oil & Gas

Analysis of multiphase
separator

– Power Generation
– Sports
Flow of lubricating
mud over drill bit Source: internet
Ashoke De (21) Computational Propulsion Laboratory, IITK
 Where is CFD used? (Power Generation)

• Where is CFD used?

– Aerospace
– Appliances
– Automotive
– Biomedical
Flow around cooling Flow in a
– Chemical Processing towers burner
– HVAC&R
– Hydraulics
– Marine
– Oil & Gas
– Power Generation
– Sports Pathlines from the inlet
Flow pattern through a water colored by temperature
turbine. during standard
Source: internet operating conditions
Ashoke De (22) Computational Propulsion Laboratory, IITK
 Where is CFD used? (Sports)

• Where is CFD used?

– Aerospace
– Appliances
– Automotive
– Biomedical
– Chemical Processing
– HVAC&R
– Hydraulics
– Marine
– Oil & Gas
– Power Generation
– Sports
Source: internet
Ashoke De (23) Computational Propulsion Laboratory, IITK
 Introduction
 What is Computational Fluid Dynamics(CFD)?
 Why and where use CFD?
 Physics of Fluid
 Navier-Stokes Equation
 Numerical Discretization
 Grids
 Boundary Conditions
 Numerical Staff

Ashoke De (24) Computational Propulsion Laboratory, IITK

 Physics of Fluid
 Fluid = Liquid + Gas
 Density ρ const incompress ible
 
variable compressib le

Viscosity μ:
resistance to flow of a fluid
 Ns 
  3 
 ( Poise)
m 
Substance Air(18ºC) Water(20ºC) Honey(20ºC)
Density(kg/m3) 1.275 1000 1446
Viscosity(P) 1.82e-4 1.002e-2 190

Ashoke De (25) Computational Propulsion Laboratory, IITK

 Physics of Fluid

Fluid Mechanics

Inviscid Viscous

Laminar Turbulence

Internal External
Compressible Incompressible (airfoil, ship)
(pipe,valve)
(air, acoustic) (water)

Components of Fluid Mechanics

Ashoke De (26) Computational Propulsion Laboratory, IITK

 Physics of Fluid
 CFD codes typically designed for representation of
specific flow phenomenon

• Viscous vs. inviscid (no viscous forces) (Re)

• Turbulent vs. laminar (Re)
• Incompressible vs. compressible (Ma)
• Single- vs. multi-phase (Ca)
• Thermal/density effects and energy equation (Pr, g, Gr, Ec)
• Free-surface flow and surface tension (Fr, We)
• Chemical reactions, mass transfer
• etc…
Ashoke De (27) Computational Propulsion Laboratory, IITK
 Introduction

 What is Computational Fluid Dynamics(CFD)?

 Why and where use CFD?
 Physics of Fluid
 Navier-Stokes Equation
 Numerical Discretization
 Grids
 Boundary Conditions
 Numerical Staff

Ashoke De (28) Computational Propulsion Laboratory, IITK

 Navier-Stokes Equations

Claude-Louis Navier George Gabriel Stokes

C.L. M. H. Navier, Memoire sur les Lois du Mouvements des Fluides, Mem. de l’Acad. d. Sci.,6, 398 (1822)
C.G. Stokes, On the Theories of the Internal Friction of Fluids in Motion, Trans. Cambridge Phys. Soc., 8, (1845)

Source: internet
Ashoke De (29) Computational Propulsion Laboratory, IITK
 Conservation law

m in m out
in M out

dM
 m in  m out
dt

m in  m out
Mass
dM Momentum
0
dt Energy

Ashoke De (30) Computational Propulsion Laboratory, IITK

 Navier-Stokes Equation I

 Mass ConservationContinuity Equation

D U i
 0 Compressible
Dt xi
D
  const, 0
Dt

U i
0 Incompressible
xi

Ashoke De (31) Computational Propulsion Laboratory, IITK

 Navier-Stokes Equation II

 Momentum ConservationMomentum Equation

U j U j P  ij
  U i    g j

t xi x j xi 
I
   V
II III IV

 U j U i  2
 ij         ij  U k
I : Local change with time  xi x j  3 xk
II : Momentum convection
III: Surface force
IV: Molecular-dependent momentum exchange(diffusion)
V: Mass force

Ashoke De (32) Computational Propulsion Laboratory, IITK

 Navier-Stokes Equation III

Momentum Equation for Incompressible Fluid

 ij   U j U i  2  U k
       ij 
xi xi  x 
x j  3 xi xk
 i
U i
0
xi  ij  2U j  2U j
 U i
    
xi xi2 x j xi xi2

U j P U j  2U j
  U i    g j
t xi x j xi
2

Ashoke De (33) Computational Propulsion Laboratory, IITK

 Navier-Stokes Equation IV

 Energy ConservationEnergy Equation

T T U i  2T U j
c  cU i  P   2   ij
 t

xi

xi xi
  
xi
I II III IV V
I : Local energy change with time
II: Convective term
III: Pressure work
IV: Heat flux(diffusion)
V: Irreversible transfer of mechanical energy into heat

Ashoke De (34) Computational Propulsion Laboratory, IITK

 Introduction
 What is Computational Fluid Dynamics(CFD)?
 Why and where use CFD?
 Physics of Fluid
 Navier-Stokes Equation
 Numerical Discretization
 Grids
 Boundary Conditions
 Numerical Staff

Ashoke De (35) Computational Propulsion Laboratory, IITK

 Discretization

Discretization
Analytical Equations Discretized Equations

 Discretization Methods
 Finite Difference
Straightforward to apply, simple, sturctured grids
 Finite Element
Any geometries
 Finite Volume
Conservation, any geometries

Ashoke De (36) Computational Propulsion Laboratory, IITK

 Discretization

Name of the Process Advantage Disadvantage

Method
Finite- The method includes Straightforwardness Not suitable to
Difference the assumption that and relative solve problems
Method the variation of the simplicity by which with increasing
(FDM) unknown to be a newcomer in the degree of physical
computed is field is able to complexity such
somewhat like a obtain solutions of as flows at higher
polynomial in x, y, or simple problems Reynolds
z so that higher numbers, flows
derivatives are around arbitrarily
unimportant. shaped bodies,
and strongly time-
dependent flows
Ashoke De (37) Computational Propulsion Laboratory, IITK
 Discretization
Name of Process Advantage Disadvantage
the
Method
Finite It finds solutions  Successful in  More complicated matrix
element at discrete spatial solid mechanics operations are required to solve the
Method regions (called applications. resulting system of equations
(FEM) elements) by
assuming that the  Their introduction  Meaningful variational
governing and ready formulations are difficult to obtain
differential acceptance in for high Reynolds number flows
equations apply to fluid mechanics
the continuum were due to  Variational principle-based FEM is
within each relative ease by limited to solutions of creeping
element. which flow flow and heat conduction problems
problems with
complicated
boundary shapes
could be modeled,
especially when
compared with
Ashoke De
FDMs.(38) Computational Propulsion Laboratory, IITK
 Discretization
Name of Process Advantage Disadvantage
the Method

Spectral The It can be easily  Their relative

Method approximation is combined with complexity in
based on standard FDMs. comparison with
expansions of standard FDMs
independent  Implementation of
variables into complex boundary
finite series of conditions appears
smooth to be a frequent
functions. source of
considerable
difficulty

Ashoke De (39) Computational Propulsion Laboratory, IITK

 Discretization
Name of Process Advantage Disadvantage
the
Method
Finite  Domain is divided into a Physical Not as
Volume number of non- soundnes straightforwa
Method overlapping control s rd as FDM
(FVM) volumes
 The differential equation
is integrated over each
control volume
 Piecewise profiles
expressing the variation
of the unknown between
the grid points are used
to evaluate the required
integrals
Ashoke De (40) Computational Propulsion Laboratory, IITK
 FVM-I
General Form of Navier-Stokes Equation

     
  U i      q    1, U j , T 
t xi  xi 
Local change with time Flux Source

Integrate over the 

Control Volume(CV) V xi dV  S   ni dS

Integral Form of Navier-Stokes Equation

    
V t dV  S  U i    xi   ni dS  V q dV


Local change Flux Over Source in CV

with time in CV the CV Surface

Ashoke De (41) Computational Propulsion Laboratory, IITK

 FVM-II
Conservation of Finite Volume Method
    
V t dV  S  U i    xi   ni dS  V q dV


A B

A B

Ashoke De (42) Computational Propulsion Laboratory, IITK

 FVM-III
UP UE
Approximation of Volume Integrals

m    dV   pV ; mu   i ui dV   PuPV
Vi Vi

Approximation of Surface Integrals ( Midpoint

Rule)
 P dV   P dS   Pk S k k  n, s, e, w
Ue Vi Si
k

Interpolation
 
U if (U  n ) e  0
Upwind U   P
e   

U E if (U  n ) e  0

xe  xP
Central U e  U E e  U P (1  e ) e 
xE  xP

Ashoke De (43) Computational Propulsion Laboratory, IITK

 Discretization of Cont. Eqn
One Control Volume
aPuP  aN uN  aS uS  aW uW  aEuE  0
Whole Domain
 a11 a12 a1l   u1  0 
a a22 a23 a2,l 1   u  0 
 21   2   
 . . . .   .  .
     
 . . . .   .  .
 ak1 . . . al 1,n 1   .  .
    
 ak 1,2 . . . al ,n   .  .
 . . . .   .  .
     
 . . . .   .  .
 an 1,n  k 1 an 1,n  2 an 1,n 1 an 1,n  u   0 
   n 1   
 an ,n k an ,n 1 ann   un  0 

Ashoke De (44) Computational Propulsion Laboratory, IITK

 Discretization of NS Eqn
 FV Discretization of Incompressible N-S Equation

Muh  0
duh
  C (uh )uh  Duh  Mqh  0
dt
Unsteady Convection Diffusion Source

 Time Discretization

duhn 1  f (uh )
n Explicit

dt  f (uhn , uhn 1 ) Implicit

Ashoke De (45) Computational Propulsion Laboratory, IITK

 Introduction
 What is Computational Fluid Dynamics(CFD)?
 Why and where use CFD?
 Physics of Fluid
 Navier-Stokes Equation
 Numerical Discretization
 Grids
 Boundary Conditions
 Numerical Staff

Ashoke De (46) Computational Propulsion Laboratory, IITK

 Grids

 Structured Grid
+ all nodes have the same number of
elements around it
– only for simple domains

 Unstructured Grid
+ for all geometries
– irregular data structure

 Block Structured Grid

Ashoke De (47) Computational Propulsion Laboratory, IITK

 Introduction
 What is Computational Fluid Dynamics(CFD)?
 Why and where use CFD?
 Physics of Fluid
 Navier-Stokes Equation
 Numerical Discretization
 Grids
 Boundary Conditions
 Numerical Staff

Ashoke De (48) Computational Propulsion Laboratory, IITK

 Boundary Conditions
 Typical Boundary Conditions
No-slip(Wall), Axisymmetric, Inlet, Outlet, Periodic

No-slip walls: u=0,v=0

Inlet ,u=c,v=0 Outlet, du/dx=0

dv/dy=0,dp/dx=0
r
v=0, dp/dr=0,du/dr=0
o x
Periodic boundary condition in
Axisymmetric
spanwise direction of an airfoil

Ashoke De (49) Computational Propulsion Laboratory, IITK

 Contents

 What is Computational Fluid Dynamics(CFD)?

 Why and where use CFD?
 Physics of Fluid
 Navier-Stokes Equation
 Numerical Discretization
 Grids
 Boundary Conditions
 Numerical Staff

Ashoke De (50) Computational Propulsion Laboratory, IITK

 Solvers and Numerical Staff
 Solvers
 Direct: Cramer’s rule, Gauss elimination, LU decomposition
 Iterative: Jacobi method, Gauss-Seidel method, SOR method

 Numerical Parameters
 Under relaxation factor, convergence limit, etc.
 Multigrid, Parallelization
 Monitor residuals (change of results between iterations)
 Number of iterations for steady flow or number of time steps for
unsteady flow
 Single/double precisions

Ashoke De (51) Computational Propulsion Laboratory, IITK

Criteria Classification
Detail of PDEsExamples
The order of a PDE is First order: ∂/∂x –G ∂/∂y= O
determined by the highest- Second order: ∂2/∂x2 - ∂/∂y=O
order order partial derivative Third order: [∂3/∂x3]2 + ∂2/∂x∂y +
present in that equation ∂/∂y = O
If the coefficients are a ∂2/∂x2 + b ∂2/∂x∂y + c ∂2/∂y2 + d
constants or functions of the = O
independent variables only,
then Eq. is linear. If
the coefficients are functions
linearity of the dependent variables
and/or any of its derivatives
of either lower or same
order, then the equation is
nonlinear.

Ashoke De (52) Computational Propulsion Laboratory, IITK

 Classification of PDEs
Linear second-order PDEs: elliptic, parabolic, and hyperbolic.
The general form of this class of equations is:
2 2 2
       
a b c d e  f  g  0
x
2
xy y
2
x y

where coefficients are either constants or functions of the independent variables only.
The three canonical forms are determined by the following criteria:

 b2 – 4ac < 0 elliptic

 b2 – 4ac = 0 parabolic
 b2 – 4ac > 0 hyperbolic

Ashoke De (53) Computational Propulsion Laboratory, IITK

 Classification of PDEs
PDE Example Explanation
Laplace’s equation: In elliptic problems, the function f(x, y) must satisfy
both, the differential equation over a closed domain
   
2 2

  0 and the boundary conditions on the closed boundary

x y of the domain.
2 2
Elliptic
Poisson’s equation:
   
2 2

  g ( x, y )
x y
2 2

Heat conduction In parabolic problems, the solution advances

Parabolic outward indefinitely from known initial values,
  
2

 always satisfying the known boundary conditions as

t x
2
the solution progresses.
Wave equation The solution domain of hyperbolic PDE has the
same open-ended nature as in parabolic PDE.
 
2 2
Hyperbolic However, two initial conditions are required to start
g
2
the solution of hyperbolic equations in contrast with
t x
2 2
parabolic equations, where only one initial condition
is required.

Ashoke De (54) Computational Propulsion Laboratory, IITK

 Classification of N-S eqn
The complete Navier–Stokes equations in three space coordinates (x, y, z)
and time (t) are a system of three nonlinear second-order equations in four
independent variables. So, the normal classification rules do not apply
directly to them. Nevertheless, they do possess properties such as
hyperbolic, parabolic, and elliptic:

• Unsteady, inviscid compressible flow. A compressible

flow can sustain sound and shock waves, and the
Hyperbolic Navier–Stokes equations are essentially hyperbolic in
Flows nature.
• For steady inviscid compressible flows, the equations
are hyperbolic if the speed is supersonic, and elliptic for
subsonic speed.

Ashoke De (55) Computational Propulsion Laboratory, IITK

 Classification of N-S eqn
Parabolic Flows Elliptic Flows Mixed Flows
•The boundary layer • The subsonic inviscid There is a possibility
flows have flow falls under this that a flow may not be
essentially parabolic category. characterized purely
character. The •If a flow has a region of by one type. For
solution marches in recirculation, information example, in a steady
the downstream may travel upstream as transonic flow, both
direction, and the well as downstream. supersonic and
numerical methods Therefore, specification of subsonic regions exist.
used for solving boundary conditions only The supersonic regions
parabolic equations at the upstream end of the are hyperbolic,
are appropriate. flow is not sufficient. The whereas subsonic
problem then becomes regions are elliptic.
elliptic in nature.

Ashoke De (56) Computational Propulsion Laboratory, IITK

 Initial and BC
The initial and boundary conditions must be specified to obtain unique numerical
solutions to PDEs:
Following Eq. depicts a problem in which the temperature within a large solid slab
having finite thickness changes in the x-direction as a function of time till steady state
(corresponding to t → ∞ ) is reached:
 T
2
T
 g
t x
2

1. Dirichlet Conditions (First Kind):

The values of the dependent variables are specified at the boundaries
in the figure:
• Boundary Conditions of first kind can be expressed as
B.C. 1 T=f (t) or T1 at x=0
t>0
B.C.2 T= T2 at x=L
• Initial Condition
T= f(x) at t= 0 0<= x =<L
or T= T0

Ashoke De (57) Computational Propulsion Laboratory, IITK

 Initial and BC
2. Neumann Conditions (Second Kind)
The derivative of the dependent variable is given as a constant or as a function of the
independent variable on one boundary:
T
 0 at  x  L  and t  0
x
This condition specifies that the temperature gradient at the right boundary is zero
(insulation condition).

Cauchy conditions: A problem that combines both Dirichlet and Neumann conditions
is considered to have Cauchy conditions:

Fig: Cauchy conditions

Ashoke De (58) Computational Propulsion Laboratory, IITK

 Initial and BC
3. Robbins Conditions (Third Kind)
The derivative of the dependent variable is given as a function of the
dependent variable on the boundary.
For the heat conduction problem, this may correspond to the case of
cooling of
a large steel slab of finite thickness “L” by water or oil, the heat transfer
coefficient h being finite:

Ashoke De (59) Computational Propulsion Laboratory, IITK

 Initial and Boundary value probs
On the basis of their initial and boundary conditions, PDEs may be further classified into
initial value or boundary value problems.

 Initial Value Problems:

In this case, at least one of the independent variables has an open region. In the unsteady
state heat conduction problem, the time variable has the range 0 ≤ t ≤ ∞ , where no
condition has been specified at t = ∞ ; therefore, this is an initial value problem.

 Boundary Value Problems:

When the region is closed for all independent variables and conditions are specified at all
boundaries, then the problem is of the boundary value type. An example of this is the
three-dimensional steady-state heat conduction (with no heat generation) problem, which
is mathematically represented by the equation:

Ashoke De (60) Computational Propulsion Laboratory, IITK

Ashoke De (61) Computational Propulsion Laboratory, IITK