PFJL Lecture 21, 1 Numerical Fluid Mechanics 2.29 2.29 Numerical Fluid Mechanics Fall 2011 – Lecture 21 REVIEW Lecture 20: Time-Marching Methods and ODEs–IVPs • Time-Marching Methods and ODEs – Initial Value Problems – Euler’s method – Taylor Series Methods • Error analysis: for two time-levels, if truncation error is of O(h n ), the global error is of O(h n-1 ) – Simple 2 nd order methods • Heun’s Predictor-Corrector and Midpoint Method (part of Runge-Kutta’s methods) • To achieve higher accuracy in time: utilize information (known values of the derivative in time, i.e. the RHS f ) at more points in time – Runge-Kutta Methods • Additional points are between t n and t n+1 – Multistep/Multipoint Methods: Adams Methods • Additional points are at past time steps – Practical CFD Methods – Implicit Nonlinear systems – Deferred-correction Approach 0 0 ( ) or ( ,) ; with ( ) d d t t dt dt Φ Φ BΦ bc BΦ Φ Φ 1 1 (,) n n t n n t ft dt
24
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2.29 Numerical Fluid Mechanics Fall 2011 Lecture 21 Fluid Mechanics . PFJL Lecture 21, ... Numerical Methods for Engineers, ... and Chapter 8 on “Complex Geometries” of “J.
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PFJL Lecture 21, 1 Numerical Fluid Mechanics 2.29
2.29 Numerical Fluid Mechanics
Fall 2011 – Lecture 21
REVIEW Lecture 20: Time-Marching Methods and ODEs–IVPs
• Time-Marching Methods and ODEs – Initial Value Problems
– Euler’s method – Taylor Series Methods
• Error analysis: for two time-levels, if truncation error is of O(hn), the global error is of O(hn-1) – Simple 2nd order methods
• Heun’s Predictor-Corrector and Midpoint Method (part of Runge-Kutta’s methods)
• To achieve higher accuracy in time: utilize information (known values of the derivative in time, i.e. the RHS f ) at more points in time – Runge-Kutta Methods
• Additional points are between tn and tn+1 – Multistep/Multipoint Methods: Adams Methods
• Additional points are at past time steps – Practical CFD Methods – Implicit Nonlinear systems – Deferred-correction Approach
0 0( ) or ( , ) ; with ( )d d t tdt dt
Φ ΦBΦ bc B Φ Φ Φ
11 ( , )
n
n
tn n
t
f t dt
PFJL Lecture 21, 2 Numerical Fluid Mechanics 2.29
TODAY (Lecture 21): End of Time-Marching
Methods, Grid Generation and Complex Geometries
• Time-Marching Methods and ODEs – IVPs: End – Multistep/Multipoint Methods – Implicit Nonlinear systems – Deferred-correction Approach
• Complex Geometries – Different types of grids – Choice of variable arrangements
• Grid Generation – Basic concepts and structured grids
• Chapters 25 and 26 of “Chapra and Canale, Numerical Methods for Engineers, 2010/2006.”
• Chapter 6 (end) and Chapter 8 on “Complex Geometries” of “J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics. Springer, NY, 3rd edition, 2002”
• Chapter 6 (end) on “Time-Marching Methods for ODE’s” of “H. Lomax, T. H. Pulliam, D.W. Zingg, Fundamentals of Computational Fluid Dynamics (Scientific Computation). Springer, 2003”
• Chapter 9 on “Grid Generation” of T. Cebeci, J. P. Shao, F. Kafyeke and E. Laurendeau, Computational Fluid Dynamics for Engineers. Springer, 2005.
• Ref on Grid Generation only:
– Thompson, J.F., Warsi Z.U.A. and C.W. Mastin, “Numerical Grid Generation, Foundations and Applications”, North Holland, 1985
PFJL Lecture 21, 4 Numerical Fluid Mechanics 2.29
Multistep/Multipoint Methods
• Additional points are at time steps at which data has already
been computed
• Adams Methods: fitting a (Lagrange) polynomial to the
derivatives at a number of points in time
– Explicit in time (up to tn): Adams-Bashforth methods
– Implicit in time (up to tn+1): Adams-Moulton methods
– Coefficients βk’s can be estimated by Taylor Tables:
• Fit Taylor series so as to cancel higher-order terms
1 ( , )n
n n kk k
k n K
f t t
11 ( , )
nn n k
k kk n K
f t t
PFJL Lecture 21, 5 Numerical Fluid Mechanics 2.29
Example: Taylor Table for the
Adams-Moulton 3-steps (4 time-nodes) Method
11 1 1 2
1 1 0 1 1 2 2
dDenoting , , ' ( , ) and ' ( , ) , one obtains for 2 :
( , ) ( , ) ( , ) ( , ) ( , )
nn n
n n n k n n n nk n k n n n n
k K
uh t u u f t u u f t u Kdt
u u f t u t h f t u f t u f t u f t u
Taylor Table: • The first row (Taylor
series) + the last 5 rows (Taylor series for each term) must sum to zero
• This can be satisfied up to the 5th column (4th order term)
• Hence, the AM method with 4-time levels is 4th order accurate
1 0 1 2solving for the ' 9 / 24, 19 / 24, 5/ 24 and 1/ 24k s
PFJL Lecture 21, 6 Numerical Fluid Mechanics 2.29
Example of Adams Methods for
Time-Integration
(Adams-Bashforth, with ABn meaning nth order AB)
(Adams-Moulton, with AMn meaning nth order AM)
PFJL Lecture 21, 7 Numerical Fluid Mechanics
2.29
Practical Time-Integration Methods for CFD
• High-resolution CFD requires large discrete state vector sizes to store the spatial
information
• This means that up to two times (one on each side of the current time step) have
often been utilized (3 time-nodes):
• Rewriting this equations in a way such that differences wrt. the Euler’s method are
easily seen, one obtains (θ = 0 for explicit schemes):
1 1 11 1 0 1 1( , ) ( , ) ( , )n n n n n
n n nu u h f t u f t u f t u
1 1 1 11 1(1 ) (1 2 ) ( , ) (1 ) ( , ) ( , )n n n n n n
• Consider the nonlinear system (discrete in space):
• For an explicit method in time, solution is straightforward
– For explicit Euler:
– More general, e.g. AB:
• For an implicit method
– For Implicit Euler:
– More general:
=> a nontrivial scheme is needed to obtain
2.29 Numerical Fluid Mechanics PFJL Lecture 21, 8
0 0( , ) ; with ( )d t tdt
Φ B Φ Φ Φ
1 ( , ) tn n n nt Φ Φ B Φ
1 1( , ,..., , ) tn n n n K nt Φ F Φ Φ Φ
1nΦ
1 1 1 1
1 1 1 1
( , , ,..., , ) t or( , , ,..., , ) 0 ; with t
n n n n n K n
n n n n K n n
tt
Φ F Φ Φ Φ ΦF Φ Φ Φ Φ F F Φ1 1 1 1( , , ,..., , ) 0 ; with tn n n n K n n1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1n n n n K n n n n n n K n n1 1 1 1n n n n K n n1 1 1 1 1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with t ( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with t1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 11 1 1 1n n n n K n n1 1 1 1 1 1 1 1n n n n K n n1 1 1 1 1 1 1 1n n n n K n n1 1 1 1 1 1 1 1n n n n K n n1 1 1 11 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1F Φ Φ Φ Φ F F Φ
n n n n K n nΦ Φ Φ Φ F F Φ
n n n n K n n1 1 1 1n n n n K n n1 1 1 1 1 1 1 1n n n n K n n1 1 1 1Φ Φ Φ Φ F F Φ
1 1 1 1n n n n K n n1 1 1 1 1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with tΦ Φ Φ Φ F F Φ( , , ,..., , ) 0 ; with t( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with tΦ Φ Φ Φ F F Φ( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with t( , , ,..., , ) 0 ; with tΦ Φ Φ Φ F F Φ( , , ,..., , ) 0 ; with t( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with tΦ Φ Φ Φ F F Φ( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with t( , , ,..., , ) 0 ; with tt( , , ,..., , ) 0 ; with tΦ Φ Φ Φ F F Φ( , , ,..., , ) 0 ; with tt( , , ,..., , ) 0 ; with t( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with tt( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with tΦ Φ Φ Φ F F Φ( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with tt( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with t( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with t ( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with tΦ Φ Φ Φ F F Φ( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with t ( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with t1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1Φ Φ Φ Φ F F Φ
1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 11 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1t1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1t1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1Φ Φ Φ Φ F F Φ
1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1t1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1t1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 Φ Φ Φ Φ F F Φ ( , , ,..., , ) 0 ; with t ( , , ,..., , ) 0 ; with tΦ Φ Φ Φ F F Φ( , , ,..., , ) 0 ; with t ( , , ,..., , ) 0 ; with tn n n n K n n n n n n K n nΦ Φ Φ Φ F F Φ
n n n n K n n n n n n K n n( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with t ( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with tΦ Φ Φ Φ F F Φ( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with t ( , , ,..., , ) 0 ; with tn n n n K n n( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1 1 1 1 1n n n n K n n1 1 1 1 1 1 1 1n n n n K n n1 1 1 1 1 1 1 1n n n n K n n1 1 1 1Φ Φ Φ Φ F F Φ
1 1 1 1n n n n K n n1 1 1 1 1 1 1 1n n n n K n n1 1 1 1 1 1 1 1n n n n K n n1 1 1 1 1 1 1 1n n n n K n n1 1 1 11 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1Φ Φ Φ Φ F F Φ
1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1 1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1n n n n K n n1 1 1 1( , , ,..., , ) 0 ; with t1 1 1 1
1 1 1( , ) tn n n nt Φ Φ B Φ
Implementation of Implicit Time-Marching Methods:
Larger dimensions and Nonlinear systems
• Two main options for an implicit method, either:
1. Linearize the RHS at tn :
• Taylor Series:
where
• Hence, the linearized system (for the frequent case of system not explicitly
function of t):
2. Use an iteration scheme at each time step, e.g. fixed point iteration (direct),
Newton-Raphson or secant method
• Newton-Raphson:
n• Iteration often rapidly convergent since initial guess to start iteration at t close n+1to unknown solution at t
2.29 Numerical Fluid Mechanics PFJL Lecture 21, 9
2 n+1( , ) = ( , ) ( ) ( ) ( ) for t tn
n n n n n nt t t t O t tt
BB Φ B Φ J Φ Φ
; i.e. [ ] (Jacobian Matrix)n
n n ii j
j
B BJ JΦ Φ
( ) + ( )n n n nd ddt dt
Φ ΦB Φ J Φ B Φ J Φ
1
1 1 1 11 1 1
1 ( ) ( , )'( )
n n n nr r r r r rn
r r
x x f x tf x
FΦ Φ F ΦΦ
1
1 1 1 11 1 1 1 1 1 1 11 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 11 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 11 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 11 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 11 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 F
F 1 1 1 1 1 1 1 1F1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1F1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 1F1 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 11 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1F1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 F
1 1 1 1( ) ( , )1 1 1 1( ) ( , )1 1 1 1( ) ( , )n n n n( ) ( , )( ) ( , )x x f x t( ) ( , )( ) ( , )n n n n( ) ( , )x x f x t( ) ( , )n n n n( ) ( , )1 1 1 1 1 1 1 11 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 11 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 11 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 11 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1( ) ( , )x x f x t( ) ( , )Φ Φ F Φ( ) ( , )x x f x t( ) ( , )( ) ( , )n n n n( ) ( , )x x f x t( ) ( , )n n n n( ) ( , )Φ Φ F Φ( ) ( , )n n n n( ) ( , )x x f x t( ) ( , )n n n n( ) ( , )1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1Φ Φ F Φ
1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 1n n n n1 1 1 1 1 1 1 1n n n n1 1 1 11 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1 1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1x x f x t1 1 1 1( ) ( , )1 1 1 1n n n n1 1 1 1( ) ( , )1 1 1 1
– Hence, for non-orthogonal grids, grid-oriented velocity components often used
• Collocated arrangements (mostly used here) – The simplest one: all variables share the same CV – Requires more interpolation
Variable arrangements on a non-orthogonal grid. Illustrated are a staggered arrangement with (i) contravarient velocity components and (ii) Cartesian velocity components, and (iii) a colocated arrangement with Cartesian velocity components.