Course 1: Fluid Mechanics and Energy Conversion
Discretization Methods
in Fluid Dynamics
Mayank Behl
B-tech. 3rd Year
Department of Chemical Engineering
Indian Institute of Technology Delhi
3/8/2007
Supervisor:
Dr. G.Biswas
Indian Institute of Technology
Kanpur
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OUTLINE
1.
2.
3.
4.
5.
6.
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Basics of PDE
Mathematical Overview of fluid flow
Need For Computational Methods
Basic Discretization Methods in use
Finite Difference Method
Fluid Flow Modeling
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Partial differential equations (PDEs) are used to
formulate and solve problems related to any phenomena
that is distributed in space and time :
„
„
„
„
„
Heat flow
∂φ
∂φ
∂ φ
∂ φ
∂ φ
A
+D
+E
+ Fφ = G ( x, y )
+B
+C
∂
x
∂
y
∂
x
∂
y
∂
x
∂
y
Fluid flow
propagation of sound
electrodynamics,
Elasticity…….
2
2
2
2
2
Most of the fluid and related transport phenomena
can be modeled using PDEs
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Classification of PDEs
∂ 2φ
∂ 2φ
∂ 2φ
∂φ
∂φ
+C 2 + D
+E
A 2 +B
+ Fφ = G ( x , y )
∂x∂y
∂x
∂y
∂x
∂y
If A,B,C,D… all constants or f(x,y)
Linear PDE
If any of A,B,C,D… contains Φ, Φ’ etc
Non-linear PDE
B2 – 4AC<0
Elliptic equation
B.V.P.
B2 –
4AC=0
Parabolic equation
I.B.V.P;
B2 – 4AC>0
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Hyperbolic equation
∂2φ
∂2φ
+
= 0
2
2
∂x
∂y
∂φ
∂ 2φ
=α
∂t
∂x 2
∂ 2φ
= β
∂t 2
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∂ 2φ
∂x 2
Irrotational flows
steady Heat Transfer
Unsteady viscous
flows
Vibration problems
4
Boundary conditions
„
„
„
Dirchlet Boundary
condition:
Neumann Boundary
condition:
Mixed Boundary
condition:
φ = φ1 (r )
Given boundary temperature
∂φ
= φ 2 (r )
∂n
Given boundary heat flux
∂φ
aφ + b
= φ3 (r )
∂n
Boundary heat flux dependent on boundary temperature
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Mathematical description of flows;
Governing Equations of Fluid Dynamics
„
Equation of continuity: a differential mass balance
∂ρu ∂ρv ∂ρw
∂ρ
+
+
=−
∂x
∂y
∂z
∂t
Δz
∂ρ
In vector notation: ∇ ⋅ [ρ u] = −
∂t
For constant density of the fluid:
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Δy
(x,y,z)
Δx
∇⋅u = 0
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Governing Equations of Fluid Dynamics (2)
∂
„Equation of motion:
ρu + ∇ ⋅ ρuu = −∇p − ∇ ⋅ τ + ρg
∂t
for incompressible, Newtonian Fluids , under laminar flow:
⎛∂
⎞
ρ ⎜ u + u ⋅ ∇u ⎟ = −∇p + μ ∇ 2 u + ρg
⎝ ∂t
⎠
Navier-Stokes Equation
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The Navier-Stokes equations
∂u
∂u ∂u ∂u
1 ∂p μ ⎛ ∂2u ∂2u ∂2u ⎞
+ ⎜⎜ 2 + 2 + 2 ⎟⎟ + gx
+u +v + w = −
∂t
∂x ∂y
∂z
ρ ∂x ρ ⎝ ∂x ∂y ∂z ⎠
1 ∂p μ ⎛ ∂2v ∂2v ∂2v ⎞
∂v
∂v ∂v ∂v
+ ⎜⎜ 2 + 2 + 2 ⎟⎟ + g y
+u +v + w = −
ρ ∂y ρ ⎝ ∂x ∂y ∂z ⎠
∂z
∂x ∂y
∂t
(Non linear
PDEs)
1 ∂p μ ⎛ ∂2 w ∂2 w ∂2 w⎞
∂w ∂w ∂w ∂w
+ ⎜⎜ 2 + 2 + 2 ⎟⎟ + gz
+u +v + w = −
ρ ∂z ρ ⎝ ∂x ∂y ∂z ⎠
∂z
∂x ∂y
∂t
Solution???
In general, Analytical Solution Not Possible
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Approaches to Fluid Dynamical Problems
1. Simplifications of the governing equations
Æ AFD: Analytical Fluid Dynamic
2. Experiments on scale models
Æ EFD: Experimental Fluid Dynamics
3. Discretize governing equations and solve by computers
Æ CFD Computational Fluid Dynamics
CFD is the simulation of fluids engineering system using
modeling and numerical methods
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Need for Discretization methods
¾ In general, no analytical solution exist for non-linear models
One of the way out is to approximate the solutions using
Numerical methods
¾ Advent of digital computer and advanced improvements in
computer resources
¾ Basis for CFD, which can provide unlimited details of results
¾Substantially more cost effective and more rapid than EFD
¾ability to study systems under hazardous conditions
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Various Discretization Techniques
¾ Finite Difference Method (focused in this lecture)
¾ Finite Volume Method
¾ Finite Element Method
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Discretization Methods
Finite Difference Method (focused in this lecture)
1.Governing equations in differential formÆ domain with gridÆ
replacing the partial derivatives by approximations in terms of node
values of the functionsÆ one algebraic equation per grid nodeÆ linear
algebraic equation system.
2. Applied to structured grids
Finite Volume Method
1. Governing equations in integral formÆ solution domain is subdivided
into a finite number of contiguous control volumesÆ conservation
equation applied to each CV.
2. Computational node locates at the centroid of each CV.
3. Applied to any type of grids, especially complex geometries
Finite Element Method
1. Solution domain is subdivided into a finite number OF ELEMENTS,
the governing equation is solved for each element and then overall
solution is obtained by “assembly”.
2. Equations are multiplied by a weight function before integrated over
the entire domain.
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Basics aspects of Discretization methods
► Consistency
A discretization method is said to be consistent if it can be
shown that the difference between PDEs and its finite
difference (FDE) vanishes as the mesh is refined.
Truncation error (TE): Difference between the discretized
equation and the exact one
O(∆x); O(∆t)
LimmeshÆ0 (TE)
0
O(∆t/∆x)
►Convergence: solution of the discretized equations
tends to the exact solution of the differential equation as the
grid spacing tends to zero.
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Basics aspects of Discretization methods
Stability : Differencing Method should not magnify the
errors that appear in the course off numerical solution
process
ε
n+1
i
n
i
ε
≤1
Conservation:
1. The numerical scheme should on both local and
global basis respect the conservation laws.
2. Automatically satisfied for control volume method,
either individual control volume or the whole domain.
3. Errors due to non-conservation are in most cases
appreciable only on relatively coarse grids, but hard to
estimate quantitatively
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Finite Difference Method
Replacing the derivatives of governing
PDE with finite, algebraic differences
quotients. It involves following steps:
„
„
„
Grid generation defining
geometric domain
Discretization of governing
equation using Taylor
series approximation.
Solution of simultaneous
algebraic equations.
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φφi,ji,j
φi-1,j
φi-1,j
φi+1,j
φi+1,j
Δy
Δx
Discrete grid points
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Finite Difference Method:grids
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
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Finite Difference,
approximation of the first derivative
►Taylor Series Expansion: Any continuous differentiable
function, in the vicinity of xi , can be expressed as a Taylor
series:
⎛ ∂Φ ⎞ ( x − xi )
Φ ( x ) = Φ ( xi ) + ( x − xi )⎜
⎟ +
2!
⎝ ∂x ⎠ i
2
⎛ ∂ 2 Φ ⎞ ( x − xi )
⎜⎜ 2 ⎟⎟ +
3!
⎝ ∂x ⎠ i
⎛ ∂Φ ⎞ Φi +1 − Φi xi +1 − xi
−
⎜
⎟ =
x
x
−
x
2
∂
⎝
⎠i
i +1
i
3
⎛ ∂ 3Φ ⎞
( x − xi )
⎜⎜ 3 ⎟⎟ + ... +
n!
⎝ ∂x ⎠ i
⎛ ∂ 2Φ ⎞ ( xi +1 − xi )
⎜⎜ 2 ⎟⎟ −
6
⎝ ∂x ⎠ i
2
n
⎛ ∂ nΦ ⎞
⎜⎜ n ⎟⎟ + H
⎝ ∂x ⎠ i
⎛ ∂ 3Φ ⎞
⎜⎜ 3 ⎟⎟ + H
⎝ ∂x ⎠ i
►Higher order derivatives are unknown and can be dropped when the
distance between grid points is small
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Xi+1- Xi is small
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►By writing Taylor series at different nodes, x i-1, x i+1, or both
x i-1 and x i+1, we can have:
order of accuracy
⎛ ∂Φ ⎞ Φ i +1 − Φ i
⎜
⎟ ≈
xi +1 − xi
⎝ ∂x ⎠i
Forward Difference Scheme-FDS
O(∆x)
⎛ ∂Φ ⎞ Φ i − Φ i −1
⎜
⎟ ≈
∂
x
xi − xi −1
⎝
⎠i
Backward Difference Scheme-BDS
O(∆x)
Φ i +1 − Φ i −1
⎛ ∂Φ ⎞
≈
⎜
⎟
xi +1 − xi −1
⎝ ∂x ⎠ i
Central Difference Scheme-CDS
O(∆x)²
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Finite Difference,
approximation of the second derivative
Geometrically, the second derivative is the slope of the line
tangent to the curve representing the first derivative.
⎛ ∂ 2Φ ⎞
⎜⎜ 2 ⎟⎟
⎝ ∂x ⎠i
⎛ ∂Φ ⎞
⎛ ∂Φ ⎞
−
⎜
⎟
⎜
⎟
⎝ ∂x ⎠i +1 ⎝ ∂x ⎠i
≈
xi +1 − xi
Estimate the outer derivative by FDS, and estimate the inner
derivatives using BDS, we get
⎛ ∂ 2 Φ ⎞ Φ i +1 ( xi − xi −1 ) + Φ i −1 ( xi +1 − xi ) − Φ i ( xi +1 − xi −1 )
⎜⎜ 2 ⎟⎟ ≈
(xi +1 − xi )2 (xi − xi −1 )
⎝ ∂x ⎠i
For equidistant spacing of the points:
⎛ ∂ 2Φ ⎞
Φ − 2Φ i + Φ i −1
⎜⎜ 2 ⎟⎟ ≈ i +1
(Δx )2
⎝ ∂x ⎠ i
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Higher-order approximations for the second derivative can be
derived by including more data points, such as xi-2, and xi+2, even
xi-3, and xi+3
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Discretization using Explicit Method
„
Consider unsteady viscous flow
∂Φ
∂ 2Φ
=α
∂t
∂x 2
Φ in +1 − Φ in
Φ n i +1 − 2Φ n i + Φ n i −1
=α
Δt
(Δx )2
The Only unkown
:Φin+1
Explicit method , values at time n+1 computed from values at time n
Advantages: solution algorithm is simple i.e.
- direct computation without solving system of algebraic equation
- few number of operations per time step
Disadvantage:
strong
conditions
on out
time the
step calculations
for stability over
Requires
many time
steps
to carry
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a
given time interval
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Implicit Method
Crank-Nicholson Method
∂u
∂ 2u
=α 2
∂t
∂x
uin+1 − uin
u n i+1 + uin++11 − 2u n i − 2uin+1 + u n i−1 + uin−+11
=α
2
Δt
2(Δx)
ƒ values at time n+1 computed using the unknown values at time n+1
ƒ Second order accurate in time and space O(∆t)2,O(∆x)2
ƒto be solved simultaneously at all the grid points as a system of
algebraic equations
ƒ Unconditionally stable
rearranging the above equation as follows….
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Ax = C
This generates a Tridiagonal Matrix system which can be solved
using Thomas algorithm.
However, for higher dimensional flows, matrix system obtained are
much more complex and require substantially more computer time
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Alternating Direction Implicit Method
(ADI) for 2-D Flows
⎛ ∂ 2u ∂ 2u ⎞
∂u
= α ⎜⎜ 2 + 2 ⎟⎟
∂t
∂y ⎠
⎝ ∂x
Each time increment is executed into two steps:
First step
uin,+j1/ 2 − uin, j
Δt / 2
⎛ uin+1, j − 2uin, j + uin−1, j uin,+j1+1/ 2 − 2uin,+j1/ 2 + uin,+j1−1/ 2 ⎞
⎟
= α⎜
+
2
2
⎜
⎟
Δy
(Δx)
⎝
⎠
Second step
uin, +j 1 − uin, +j 1/ 2
Δt / 2
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⎛ uin++11, j − 2uin, +j 1 + uin−+11, j uin, +j +11/ 2 − 2uin, +j 1/ 2 + uin, +j −11/ 2 ⎞
⎟
= α⎜
+
2
2
⎜
⎟
Δy
(Δx )
⎝
⎠
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First direction Y
Second direction X
ADI method results in Tridiagonal Equations (for 2-d flow systems)
if it is applied along the dimension that is implicit. Thus on first step it is
applied along Y axis and on second step along on X axis
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Implicit method
„
Advantages:
Stability can be maintained over larger time
steps
„
Disadvantages
„ More
computation time per time step, every time
step requires solving a system of equations.
„ Truncation error is often large , since larger
time steps are employed.
„ Overall, a more involved procedure
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Error and Stability analysis
„
Numerical solution is influenced by following:
Discretization errors or Truncation errors:
Analytical Solution (A) – exact Finite Difference Solution (D)
=A–D
Computational Round-Off Error (e)
ε=Numerical Solution (N) – exact Finite Difference Solution (D)
N=D+ε
By definition, N,D follow the Discretized Equation
Thus, ε should also follow the equation:
ε in +1 − ε in
Δt
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=α
ε n i +1 − 2ε n i + ε n i −1
(Δx )2
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Errors and Stability Analysis
(Cont.)
„
Substituting the value
of εm(x,t) in the F.D.E.,
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Stability: Differencing Method should not magnify the errors that
appear in the course off numerical solution process
ε in +1
≤1
n
εi
For stability:
Von Neumann Stability Analysis :
assuming error distribution follows Fourier series in space domain and
exponential in time
ε ( x, t ) = ε at ∑ ε
ik m x
m
Putting an individual term in FD equation and using the above stability
criteria
α (Δt )
1
≤
2
2
(Δx)
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Fluid Flow Modeling Using
Finite Difference Method
„
Nonlinear Fluid Flow equations involve:
time dependent multiple variables
Convective terms
Diffusive terms
„
Sample fluid flow equation : Burger’s Equation
⎛ ∂ 2ζ
∂u
∂ζ
+u
= ν ⎜⎜ 2
∂t
∂x
⎝ ∂x
„
Inviscid Burger’s Equation
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⎞
⎟⎟
⎠
∂ζ
∂u
+u
=0
∂t
∂x
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Properties of Fluid Flow Equations
Effect of Finite Difference method on properties of
fluid flow equations
„
Conservative Property: To maintain the integral conservation
relation of continuum in finite difference representation.
It depends on the form of continuum equation being
discretized.
∂
∂ζ
Conservative form:
= − (uζ )
∂t
Non-conservative form:
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∂x
∂ζ
∂ζ
= −u
∂t
∂x
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Integrating the
CONSERVATIVE FORM
ζ in +1 − ζ in
∫
Δt
I2
∑ζ
i = I1
n +1
i
Δx = α ∫
I2
Δx − ∑ ζ Δx
i = I1
Δt
(uζ ) n i +1 − (uζ ) n i −1
Δx
2Δx
I2
n
i
=
n
n
n
n
(
u
ζ
−
u
ζ
∑ i +1 i +1 i −1 i −1 )
i = I1
2
= (uζ ) I1 −1 / 2 − (uζ ) I 2 +1 / 2
I1
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I2
32
Week Conservative Form in Fluid Flow
„
Conservative form of advective part is of prime importance for
fluid flow modeling. Navier Stokes in week conservative form:
∂u ∂u2 ∂uv ∂uw 1 ∂p μ ⎛ ∂2u ∂2u ∂2u ⎞
+
=−
+ ⎜⎜ 2 + 2 + 2 ⎟⎟ + gx
+
+
∂t ∂x ∂y ∂z
ρ ∂x ρ ⎝ ∂x ∂y ∂z ⎠
∂v ∂uv ∂v2 ∂vw 1 ∂p μ ⎛ ∂2v ∂2v ∂2v ⎞
=−
+ ⎜⎜ 2 + 2 + 2 ⎟⎟ + g y
+
+
+
∂t ∂x ∂y ∂z
ρ ∂y ρ ⎝ ∂x ∂y ∂z ⎠
1 ∂p μ ⎛ ∂2 w ∂2w ∂2 w⎞
∂w ∂uw ∂vw ∂w2
+ ⎜⎜ 2 + 2 + 2 ⎟⎟ + gz
=−
+
+
+
ρ ∂z ρ ⎝ ∂x ∂y ∂z ⎠
∂t ∂x ∂y ∂z
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Discretizing Fluid Flow equations using
Upwind Scheme
„
Upwind Scheme of discretization is necessary for convection
dominated flows
∂ζ
∂ζ
= −u
∂t
Backward Differencing Scheme
ζ in +1 − ζ in
Δt
ζ in +1 − ζ in
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for u>0
uζ n i − uζ n i −1
=−
Δx
Forward Differencing Scheme
Δt
∂x
for u<0
uζ n i +1 − uζ n i
=−
Δx
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Why use Upwind scheme
„
Upwind Scheme of discretization is necessary for convection
dominated flows for obtaining numerically stable results
Central Difference Scheme applied to highly convective flows
give unconditionally Unstable Results
CDS:
FDS:
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ζ in +1 − ζ in
uζ n i +1 − uζ n i −1
=0
+
2Δxn
Δtn
n +1
n
ζ i − ζ i uζ i +1 − uζ i
= 0, u > 0
+
Δt
Δx
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Violates
Von Neumann
Stability
Criteria
35
Why use Upwind scheme
For unsteady, convection-diffusion eqn.
ζ
CDS:
−ζ
uζ − uζ
+
Δt
2Δx
n +1
i
n
n
i
i +1
n
i −1
=ν
ζ − 2ζ + ζ
n
n
i +1
i
Δx
n
i −1
2
∂ζ
⎛
⎛ u Δt ⎞ ⎞
+ uζ = ⎜ν − ⎜
⎟ ⎟ζ
∂t
⎝ 2 ⎠⎠
⎝
2
x
xx
⎛
⎛ u Δt ⎞ ⎞
⎜ν − ⎜
⎟⎟ ≥ 0
⎝ 2 ⎠⎠
⎝
2
⎛ 1 u Δt ⎞ uΔx ⎞
ν ⎜1 − ⎛⎜
⎟≥0
⎟
⎝ 2 ⎝ Δx ⎠ ν ⎠
ν ⎛⎜1 − C * Re ⎞⎟ ≥ 0
⎝
1
2
Re ≤
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⎠
2
C
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⎛ uΔx ⎞
Re = ⎜
⎟
⎝ ν ⎠
Δx
CELL Reynold’s No.
36
Upwind Scheme and Transportive Property
∂ζ
∂
(u ζ
= −
∂x
∂t
using upwind scheme (u > 0) ⇒
)
ζ in+1 − ζ in
Δt
uζ n i − uζ n i −1
=−
Δx
let a perturbati on δ in ζ occurs at m th location
• Then for at a downstream location (m + 1)
ζ mn ++11 − ζ mn +1
0 − uδ
uδ
=+
At (m+1) location
Δt
Δx
Δx
which follows the rationale for the transport ive proprerty
=−
• At (m) location
ζ mn +1 − ζ mn
uδ − 0
uδ
At (m) location
=−
Δt
Δx
Δx
perturbati on being transport ed out of the region
=−
• At (m - 1) location
ζ mn +−11 − ζ mn −1
0−0
=0
Δt
Δx
no perturbation carried upstream
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=−
At (m-1) location
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Problems of Using Forward Time Central Space method (FTCS)
At (m - 1) location
ζ mn −+11 − ζ mn −1
Δt
=−
uδ − 0
uδ
=−
≠0
2Δx
2Δx
Which indicates that Transportive Property is violated
Hence, only upwind method maintains unidirectional flow
of information.
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Upwind Method and Artificial Viscosity(1/2)
⎛ ∂ 2ζ
∂u
∂ζ
+u
= ν ⎜⎜ 2
∂t
∂x
⎝ ∂x
ζ in +1 − ζ in
Δt
ζ in +1 − ζ in
Δt
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⎞
⎟⎟
⎠
uζ n i − uζ n i −1
=−
Δx
uζ n i +1 − uζ n i
=−
Δx
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Upwind Method and Artificial Viscosity (2/2)
<1
For algorithm’s stability, artificial viscosity will necessarily be present
Source of inaccuracy
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ve > 0
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Upwind Differencing Method
Advantages and Limitations
„
Upwind effect necessary to maintain transportive property of flow
equations
„
Although space centered differences (CDS) are more accurate than
upwind scheme (as indicated by Taylor Series expansion), the whole
system is more realistically modeled using Upwind Differencing Scheme.
‰
Leads to Artificial viscosity or “false diffusion” which is a cause of
inaccuracy.
v >0
e
Plausible Improvement: Higher order Upwind Method
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Second Upwind Differencing Scheme
Let us consider
where
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Considering momentum equation
ζ ≡u
Here a factor η is introduced which expresses a
weighted average of central and upwind differencing
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Here 0<η<1
For η=0, above equation becomes centered in space
For η=1, above equation follows full upwind
Hence, η brings about an upwind bias in the difference quotient
Accuracy of the differencing scheme can be adjusted using η value
Second upwind formulation possesses both
the conservative and transportive property
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Summary
„
„
„
„
„
Finite Difference methods discretizes the differential form of governing
equation using Taylor Series expansion.
The partial derivatives of governing PDE are replaced with finite,
algebraic differences quotients at the corresponding nodes.
Leads to linear algebraic equation system, with one algebraic equation
per grid node.
Best Suited for Structured Grids only.
Flow equation properties can be maintained using simple schemes
like upwind differencing.
Discretization methods provide more cost effective and more rapid
approach for solving Fluid Flow problems as compared to
experimental methods
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Summary
Finite Difference Approach
FD Approaches Can Be Divided Into Various
Categories Depending Upon Type of PDE Being
Solved:
1.Elliptic PDE: Algebraic System, Gauss Siedel
2.One-dimensional Parabolic PDE
A) Explicit
B) Implicit
3. Two-dimensional Parabolic PDE
ADI Method
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Acknowledgement
„
This lecture has been inspired by the
study material provided by Prof. G.Biswas
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References
„
Chapra and Canale, “Numerical Methods for
Engineers”,4th Ed., Tata McGraw-Hill.
„
Som and Biswas, “Introduction to Fluid Mechanics and
Fluid Machines”, Tata McGraw-Hill
„
Anderson.J; “Computational Fluid Dynamics”
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Thank you !
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