Workshop on Iterative Errors in Unsteady Flow Simulations

Workshop on Iterative Errors inUnsteady Flow Simulations

L. Eça, G. Vaz and M. Hoekstra

3-5 May2017

Contents• Objective

• Test Case- Definition of flow problem

- Quantities of interest

• Participants

• Overview of Codes andSolution Strategies

• Submitted Results

• Observations and Future Work

Workshop on Iterative Errors in Unsteady Flow Simulations

ASME V&V Symposium, 3-5 May 2017

Objective

• Investigate the influence of iterative errors on

unsteady flow simulations performed with flow

solvers using implicit time integration

─ Check if different flow solvers exhibit the

the same trends.

─ Create awareness to the problem.

─ Use the available data to develop iterative

error estimators that can be used in the

so-called “practical applications” where

tight iterative convergence criteria may

become too expensive.



Test Case

• Flow around a circular cylinder with Reynolds

number equal to 100 or 150. Two-dimensional

flow of an incompressible fluid.

(Nearly DNS ☺!)

�� =��

�

• Fixed computational domain and boundary

conditions.

• Four multiblock structured grids available in

different formats (CGNS, MSH, CFX,

OpenFoam).



Test Case



0=

= ∞

y

x

V

VV

0=p

0

0;0

=∂

∂

==

n

yx

y

p

VV

0;0

0

=∂

∂=

∂

∂

=

y

p

y

V

V

x

y

Test Case



Number of cells and grid refinement ratio of the set

Grid Number of cells Number of Faces on cylinder surface �

�

1 503640 1920 1

2 282174 1440 1.33

3 196266 1200 1.6

4 125910 960 2

Test Case

• Quantities of interest:

─ Time-averaged (CD )avg and standard

deviation (CD )std of the drag coefficient CD

─ Standard deviation (CL )std of the lift

coefficient CL

─ Strouhal number St

─ Time-averaged base pressure coefficient,

(Cpb)avg

─ Time-averaged flow separation point,

(θsep)avg



Participants

• A. Jones (Rolls-Royce PLC)

• S. Guda, I. Çelik (West Virgina University)

• Y. Chen, K.J. Maki, H. Ye (University of Michigan)

• Sanjeeb Pal (ASME member)

• L. Eça, G.Vaz, M.Hoekstra (MARIN/IST)



Overview of Codes

• ANSYS Fluent

• CFX 14.5

• OpenFoam (pimpleFoam)

• ReFRESCO 2.1

• SATURNE



Solution Strategies

• Pressure-velocity coupling

- Fully-coupled

- SIMPLE, SIMPLEC

- PISO/SIMPLE

- PISO

• Time discretization

- First-order backward differencing

- Second-order backward differencing

• Space discretization

- Second-order diffusion

- Convection: Linear upwind (β=1);

Second-order upwind

Central differences



Solution Strategies

• Iterative convergence criteria at each

time step:

- Normalized residuals corresponding to a

solution change in a simple Jacobi iteration

(Maximum or RMS values)

- Average normalized residuals (absolute value)

- Global sum of squares of cells residuals

- Maximum number of iterations



Submitted Results

• Re=100

- PISO; Second order upwind for convection;

Second-order backwards for time.

Normalized absolute residual (average value) with

a maximum of 100 iterations (L2a).

Grids 3 and 4 with ∆t=0.01

- SIMPLE; Second order upwind for convection;


Normalized residual (maximum value, L∞)

Grids 1, 2, 3 and 4 with several ∆t



Submitted Results

• Re=100


First-order backwards for time.


Grid 4 with ∆t =0.05

- SIMPLEC; Central differences for convection


Global sum of squares of cells residuals (L2b)

Grids 1, 2, 3 and 4 with ∆t =0.01 to ∆t =0.02



Submitted Results• Re=100, Grid 4

Time-averaged drag coefficient



Residual

(CD) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1001.26

1.27

1.28

1.29

1.3

1.31

1.32

1.33

1.34L

2a,1

st,0.01

L2b,1

st,0.02

L∞,2

nd,0.02

L∞,2

nd,0.005

L∞,2

nd,0.05

L∞,1

st,0.05

Norm, Time, ∆t





Residual

(CD) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1001.26

1.27

1.28

1.29

1.3

1.31

1.32

1.33

1.34L

2a,1

st,0.01

L2b,1

st,0.016

L∞,2

nd,0.016

L∞,2

nd,0.004

L∞,2

nd,0.04

Norm, Time, ∆t





Residual

(CD) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1001.26

1.27

1.28

1.29

1.3

1.31

1.32

1.33

1.34L

2b,1

st,0.013

L∞,2

nd,0.013

L∞,2

nd,0.0033

L∞,2

nd,0.033

Norm, Time, ∆t





Residual

(CD) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1001.26

1.27

1.28

1.29

1.3

1.31

1.32

1.33

1.34L

2b,1

st,0.01

L∞,2

nd,0.01

L∞,2

nd,0.0025

L∞,2

nd,0.025

Norm, Time, ∆t


Standard deviation drag coefficient



Residual

(CD) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.002

0.004

0.006

0.008

L2a,1

st,0.01

L2b,1

st,0.02

L∞,2

nd,0.02

L∞,2

nd,0.005

L∞,2

nd,0.05

L∞,1

st,0.05

Norm, Time, ∆t





Residual

(CD) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.002

0.004

0.006

0.008

L2a,1

st,0.01

L2b,1

st,0.016

L∞,2

nd,0.016

L∞,2

nd,0.004

L∞,2

nd,0.04

Norm, Time, ∆t





Residual

(CD) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.002

0.004

0.006

0.008

L2b,1

st,0.013

L∞,2

nd,0.013

L∞,2

nd,0.0033

L∞,2

nd,0.033

Norm, Time, ∆t





Residual

(CD) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.002

0.004

0.006

0.008

L2b,1

st,0.01

L∞,2

nd,0.01

L∞,2

nd,0.0025

L∞,2

nd,0.025

Norm, Time, ∆t


Standard deviation lift coefficient



Residual

(CL) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.1

0.15

0.2

0.25

0.3

L2a,1

st,0.01

L2b,1

st,0.02

L∞,2

nd,0.02

L∞,2

nd,0.005

L∞,2

nd,0.05

L∞,1

st,0.05

Norm, Time, ∆t





Residual

(CL) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.1

0.15

0.2

0.25

0.3

L2a,1

st,0.01

L2b,1

st,0.016

L∞,2

nd,0.016

L∞,2

nd,0.004

L∞,2

nd,0.04

Norm, Time, ∆t





Residual

(CL) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.1

0.15

0.2

0.25

0.3

L2b,1

st,0.013

L∞,2

nd,0.013

L∞,2

nd,0.0033

L∞,2

nd,0.033

Norm, Time, ∆t





Residual

(CL) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.1

0.15

0.2

0.25

0.3

L2b,1

st,0.01

L∞,2

nd,0.01

L∞,2

nd,0.0025

L∞,2

nd,0.025

Norm, Time, ∆t


Strouhal number



Residual

St

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.1

0.125

0.15

0.175

L2a,1

st,0.01

L2b,1

st,0.02

L∞,2

nd,0.02

L∞,2

nd,0.005

L∞,2

nd,0.05

L∞,1

st,0.05

Norm, Time, ∆t


Strouhal number



Residual

St

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.1

0.125

0.15

0.175

L2a,1

st,0.01

L2b,1

st,0.016

L∞,2

nd,0.016

L∞,2

nd,0.004

L∞,2

nd,0.04

Norm, Time, ∆t


Strouhal number



Residual

St

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.1

0.125

0.15

0.175

L2b,1

st,0.013

L∞,2

nd,0.013

L∞,2

nd,0.0033

L∞,2

nd,0.033

Norm, Time, ∆t


Strouhal number



Residual

St

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.1

0.125

0.15

0.175

L2b,1

st,0.01

L∞,2

nd,0.01

L∞,2

nd,0.0025

L∞,2

nd,0.025

Norm, Time, ∆t


Time-averaged base pressure coefficient



Residual

(Cp

b) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100-0.75

-0.7

-0.65

-0.6

-0.55

L2a,1

st,0.01

L∞,2

nd,0.02

L∞,2

nd,0.005

L∞,2

nd,0.05

L∞,1

st,0.05

Norm, Time, ∆t





Residual

(Cp

b) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100-0.75

-0.7

-0.65

-0.6

-0.55

L2a,1

st,0.01

L∞,2

nd,0.016

L∞,2

nd,0.004

L∞,2

nd,0.04

Norm, Time, ∆t





Residual

(Cp

b) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100-0.75

-0.7

-0.65

-0.6

-0.55

L∞,2

nd,0.013

L∞,2

nd,0.0033

L∞,2

nd,0.033

Norm, Time, ∆t





Residual

(Cp

b) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100-0.75

-0.7

-0.65

-0.6

-0.55

L∞,2

nd,0.01

L∞,2

nd,0.0025

L∞,2

nd,0.025

Norm, Time, ∆t


Time-averaged separation point



Residual

(θsep)avg

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100116

116.25

116.5

116.75

117

L∞,2

nd,0.02

L∞,2

nd,0.005

L∞,2

nd,0.05

L∞,1

st,0.05

Norm, Time, ∆t





Residual

(θsep)avg

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100116

116.25

116.5

116.75

117

L∞,2

nd,0.016

L∞,2

nd,0.004

L∞,2

nd,0.04

Norm, Time, ∆t





Residual

(θsep)avg

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100116

116.25

116.5

116.75

117

L∞,2

nd,0.013

L∞,2

nd,0.0033

L∞,2

nd,0.033

Norm, Time, ∆t





Residual

(θsep)avg

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100116

116.25

116.5

116.75

117

L∞,2

nd,0.01

L∞,2

nd,0.0025

L∞,2

nd,0.025

Norm, Time, ∆t




Tolit

L2/L

∞(φ

)

Nu

mb

erof

iter

ati

on

s

10-6

10-5

10-4

10-3

10-2

0.05

0.1

0.15

0.2

0

50

100

150

Niter

Vx

Vy

Cp

Grid 4, Courantmax

=1.6

Influence of iterative convergence criteria at each time

step (Tolit) on the number of iterations performed (Niter)and ratio between L2 and L

∞norms of the residuals




Tolit

L2/L

∞(φ

)

Nu

mb

erof

iter

ati

on

s

10-8

10-7

10-6

10-5

10-4

10-3

10-2

0.05

0.1

0.15

0.2

0

50

100

150

Niter

Vx

Vy

Cp

Grid 4, Courantmax

=6.2







Tolit

L2/L

∞(φ

)

Nu

mb

erof

iter

ati

on

s

10-6

10-5

10-4

10-3

10-2

0.05

0.1

0.15

0.2

0

50

100

150

Niter

Vx

Vy

Cp

Grid 4, Courantmax

=15.4




Submitted Results• Re=100

• Estimation of the discretization error of the

Time-averaged drag coefficient using data

from three different levels of Tolit.



ri

(CD) a

vg

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21.29

1.3

1.31

1.32

1.33

1.34

1.35Tol

it=10

-3

p=2

Tolit=10

-4

p=2

Tolit=10

-6

p= 1.7

Submitted Results

• Re=150

- Fully-coupled solution; Linear upwind for convection;


RMS of normalized residuals and “Conservation

Imbalance” (CI) with a maximum of 10 iterations per

time step (L2a, L2c).

Grids 1 (finest) and 4 (coarsest) with ∆t=0.1

- PISO/SIMPLE; Linear upwind for convection;


Maximum number of iterations (Nit).

Grid 4 with maximum Courant numbers of 1, 5

and 10



Submitted Results

• Re=150

- PISO; Second order upwind for convection;


Normalized absolute residual (average value) with

a maximum of 100 iterations (L2b).

Grids 3 and 4 with ∆t=0.01




Grids 1, 2, 3 and 4 with ∆t=0.01 to ∆t=0.02







Residual

Number of iterations

(CD) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

01234567

1.2

1.225

1.25

1.275

1.3

1.325

1.35

1.375

1.4

L2a,2

nd,0.1

L2c,2

nd,0.1

Nit,2

nd,0.003

Nit,2

nd,0.016

Nit,2

nd,0.03

L″b,1

st,0.01

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(CD) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1001.2

1.225

1.25

1.275

1.3

1.325

1.35

1.375

1.4

L″b,1

st,0.01

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(CD) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1001.2

1.225

1.25

1.275

1.3

1.325

1.35

1.375

1.4

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(CD) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1001.2

1.225

1.25

1.275

1.3

1.325

1.35

1.375

1.4

L2c,2

nd,0.1

L∞,2

nd,0.02

Norm, Time, ∆t





Residual


(CD) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

01234567

0.01

0.02

0.03

0.04

0.05

L2a,2

nd,0.1

L2c,2

nd,0.1

Nit,2

nd,0.003

Nit,2

nd,0.016

Nit,2

nd,0.03

L″b,1

st,0.01

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(CD) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0.01

0.02

0.03

0.04

0.05

L″b,1

st,0.01

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(CD) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0.01

0.02

0.03

0.04

0.05

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(CD) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0.01

0.02

0.03

0.04

0.05

L2c,2

nd,0.1

L∞,2

nd,0.02

Norm, Time, ∆t





Residual


(CL) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

01234567

0.2

0.3

0.4

0.5

0.6

L2a,2

nd,0.1

L2c,2

nd,0.1

Nit,2

nd,0.003

Nit,2

nd,0.016

Nit,2

nd,0.03

L″b,1

st,0.01

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(CL) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.2

0.3

0.4

0.5

0.6

L″b,1

st,0.01

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(CL) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.2

0.3

0.4

0.5

0.6

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(CL) st

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.2

0.3

0.4

0.5

0.6

L2c,2

nd,0.1

L∞,2

nd,0.02

Norm, Time, ∆t


Strouhal number



Residual


St

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

01234567

0.15

0.16

0.17

0.18

0.19

L2a,2

nd,0.1

L2c,2

nd,0.1

Nit,2

nd,0.003

Nit,2

nd,0.016

Nit,2

nd,0.03

L″b,1

st,0.01

L∞,2

nd,0.02

Norm, Time, ∆t


Strouhal number



Residual

St

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.15

0.16

0.17

0.18

0.19

L″b,1

st,0.01

L∞,2

nd,0.02

Norm, Time, ∆t


Strouhal number



Residual

St

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.15

0.16

0.17

0.18

0.19

L∞,2

nd,0.02

Norm, Time, ∆t


Strouhal number



Residual

St

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1000.15

0.16

0.17

0.18

0.19

L2c,2

nd,0.1

L∞,2

nd,0.02

Norm, Time, ∆t





Residual


(Cp

b) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

01234567

-1

-0.9

-0.8

-0.7

L2a,2

nd,0.1

L2c,2

nd,0.1

Nit,2

nd,0.003

Nit,2

nd,0.016

Nit,2

nd,0.03

L″b,1

st,0.01

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(Cp

b) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100-1

-0.9

-0.8

-0.7

L″b,1

st,0.01

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(Cp

b) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100-1

-0.9

-0.8

-0.7

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(Cp

b) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100-1

-0.9

-0.8

-0.7

L2c,2

nd,0.1

L∞,2

nd,0.02

Norm, Time, ∆t





Residual


(θse

p) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

01234567

105

110

115

120

L2a,2

nd,0.1

L2c,2

nd,0.1

Nit,2

nd,0.003

Nit,2

nd,0.016

Nit,2

nd,0.03

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(θse

p) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100105

110

115

120

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(θse

p) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100105

110

115

120

L∞,2

nd,0.02

Norm, Time, ∆t





Residual

(θse

p) av

g

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100105

110

115

120

L2c,2

nd,0.1

L∞,2

nd,0.02

Norm, Time, ∆t

Observations

• Convergence criteria applied at each time

step of flow solvers that use implicit time

integration may have a significant influence

on the numerical accuracy of the solution.

• Obtaining statistical convergence (periodic

solution) and observed orders of grid/time

convergence matching the formal order of the

discretization does not guarantee that the

iterative error is negligible.



Observations

• Results submitted to the Workshop use several

different convergence criteria at each time step,

distinct accuracies of time discretization and/or

various strategies to define the time step.

Therefore, comparison between different data

of different codes is troublesome.

• However, submitted data suggest that convergence

criteria that guarantee a negligible influence of

the iterative error depend on the grid refinement,

time step (Courant number) and accuracy of the

time integration technique.



Future Work

• A second edition of the Workshop will be

proposed for the V&V Symposium of 2018.

Test case (and grids) remain identical, but

simulations will be performed only at Re=100.

• Quantities of interest remain the same and

the number of levels of the iterative convergence

criteria at each time step should be preferably

larger or equal than 3, discarding conditions that

lead to solutions without vortex shedding or with an

unreasonable value of the Strouhal number.

• Three different time steps (or maximum Courant

numbers) will be prescribed.



Future Work

• Participants will be requested to report the

following information:

- Average and maximum values of the residuals

at each time step;

- Average and maximum number of iterations

performed at each time step;

- Procedure adopted to normalize the residuals;

- Number of cycles used to obtain the quantities

of interest and maximum changes obtained in

these cycles.



Acknowledgement

• We would like to thank all the groups that

submitted data and to all the participants

that attended this Workshop.

• The use of the computing resources of the

Laboratory for Advanced Computing at

University of Coimbra (http://www.lca.uc.pt)

in the preparation of this Workshop is gratefully

acknowledge.



Documents

Workshop on Iterative Errors in Unsteady Flow Simulations