Conventional symmetric filter stencil :

ˆ j jA u B u

ˆ ju

ju

Where,

Filtered variable

Unfiltered variable

M Order of the filter

Free parameter with 0.5 0.5

1 10

ˆ ˆ ˆ2

Mn

j j j j n j nn

au u u u u

Introduction toexplicit filters

Gaitonde et al. 1999

Central Second Order Filter

Taylor series expansion provides

LHS =

RHS =

Consistency demands coefficients of

should be matched.

11 1 0 1 1

ˆ ˆˆ2j j j j j j

aU U a U U U U

2 4''ˆ ˆ ˆ(1 2 ) 2 ( ..............)

2! 4!iv

j j j

h hU U U

2''1

0 2 2 ................2 2!j j j

a ha U U U

ˆj jU and U

0 11 2 ( )a a A

The second condition is obtained by fixing the Transfer function at the Nyquist limit.

T.F. = 0 at kh = in

Solve for in terms of .

Central Second Order Filter (Cont.)

0 1 cos( )

1 2 cos( )

a a khTF

kh

0 1a and a

0 1

1

2a a

At a time instant tn the Fourier-Laplace representation of filtered and unfiltered form of the unknown can be defined as,

^ ^

( , ) ( , ) likxl n nu x t U k t e dk

( , ) ( , ) likxl n nu x t U k t e dk

^( ) ( )

1 1

( , ) ( , )l j l j

N Nik x x ik x x

jl n jl nl l

a e U k t b e U k t

For the jth node, one can equate the same wave number component on either side of the filter stencil to obtain,

So the transfer function for jth node, for the filter can be written in the spectral plane as,

( )

1

( )

1

( )

l j

l j

Nik x x

jll

j Nik x x

jll

b eT k

a e

Transfer Function Definition

Properties of Filter (Transfer Function)Real part of a Transfer Function for different order

periodic filters.

Realpart

Imaginarypart

Transfer Function Variation At Different Nodes for the Sixth Order Non-Periodic Filter

Transfer function overshoot for near boundary nodes

Numerical instabilityat nodes 2 and 3

Excessive dissipationat nodes N-1 and N-2

Imaginary parts affectdispersion properties

T.F

. (R

eal)

T.F

.

( Im

agin

ary)

Real part of the Transfer Function for the Composite Filter **

At nodes 2 and N-1 : Second order filter stencil3 and N-2 : Fourth order filter stencil4 to N-3 : Sixth order filter stencil

** LOC approach – Chu & Fan (JCP 1998) , Gaitonde & Visbal (AIAA 2000)

We adopt an equation that models convection process, has exact solution and some special properties:

(1) The solution propagates the initial condition at the speed c- without any attenuation.

(2) The initial solution does not disperse.

The space-time dependent solution can be written via Fourier- Laplace transform by,

Here, k is wave number.

Characterizing Convection Dominated Flows

)(),(),( khdetkhUtxu mikxm

(1) 0

x

uc

t

u

Essential Properties of Numerical Schemes:Amplification factor ‘G’ [for CD2-Euler scheme]

02

111

h

uuc

t

uu nm

nm

nm

nm

1( , ) ( ) (B)

( , )

n

n

U kh tDefine G kh

U kh t

(C) Number CFL - h

tcNc

(D) 2 11

1 nm

nm

nm

nm uu

Ncuu

( , ) ( , ) ( )mikxn nmu x t U kh t e d kh A

For the scheme in (D) we get

sin1),( khiNNkhG cc

• By itself, this is an Unstable Scheme.

• Later, we will show how this can be made stable/ neutrally stable by ADDITIONAL EXPLICIT FILTER.

Modification of Gby Application of Explicit Filter

When a filter is applied on , to obtain the

filtered variable - this creates an equivalent amplification

factor for the jth node given by,

1( , )j nU x t

^

1( , )j nU x t

( )jT k h is the transfer function of the applied explicit filter for the jth node.

^1 1 1^

1

ˆ( , ) ( , ) ( , )( , )

( , ) ( , ) ( , )

( ) ( , )

n n n

j c n n n

j j c

U k h t U k h t U k h tG k h N

U k h t U k h t U k h t

T k h G k h N

Numerical Stabilization Property of the Filters

Initial condition

Solution at t =30

Solution of 1D wave equation by unconditionally unstable CD2 spatial discretization and Euler time discretization. Application of the given filter stabilize the solution.

Kh = 0.2πNc = 0.1C = 0.1

Numerical Properties for the Solution of Equation (1)

kh

kh

Nc Nc

OUCS3 Scheme

+ RK4

CD2 Scheme

+ RK4

The hatched portion in group velocity contours indicate “- ve” group velocity. Disturbances

at this wave number region have spurious upstream motion and are termed as “q-waves”

Comparison of Numerical Amplification Factor Contours, With and Without Applying

Filter

Scaled numerical amplification contours for the solution of 1D wave equation, when the spatial discretization is carried out by OUCS3 and time discretization is obtained by RK4.

Without Filter0.45 , 6th Order Filter

Flow Around a Cylinder Performing Rotary Oscillation.

6/1/2009 33

Examples of Unsteady Flows:Rotary Oscillation of a cylinder

Cylinder performing rotary oscillation

Amplitude:

Frequency:

Numerical Stabilization Property of the Filters

Recommended central filters:1) 4th order filter with value very close to 0.5

2) 2nd order filter apply infrequently

Effect of Frequency of Filtering on the Computed Solution

1/ˆ ( , ) ( , )( ( )) nj c j c jG kh N G kh N T kh

n: Filtering interval

This procedure can be used for any order filter in any directions.

Effect of Direction of Filtering on the Computed Solution

• Azimuthal filters smears vorticity in that direction and are not preferred over the full domain. Although, it is acceptable near the boundary.

• In the rest of the domain, use radial filters (could be central or upwinded variety).

t = 99.69

t = 100.99

t = 102.29

t = 100.99

t = 99.69

t = 102.29

Experimental flow visualization picture (Thiria et. al. (2006))

Unfiltered solution

Filtered solution6th order azimuthal filter for30 lines, = 0.496th order radial filter forcomplete domain , = 0.45

Re = 150, A =2 ,ff/f0 = 1.5

Upwind filterstencil

ˆ ju

ju

Where,

Filtered variable

Unfiltered variable

M Order of the filter

Free parameter with

Upwind coefficient

0.5 0.5

3

1 1 4 3 2 1 10

ˆ ˆ ˆ ) 5 10 10 52

nj j j j n j n j j j j j j

n

au u u u u u u u u u u

Fourth orderdissipation term

Transfer Function Variation At Different Nodes for the

Given Filter Coefficients

T.F

. (R

eal) Real Part

Imaginary Part

T.F

. (I

mag

inar

y)

T.F

. (R

eal)

T.F

. (R

eal)

Comparison of Real Part of Transfer Function, for Different Upwind Coefficients

Transfer functions are plotted for interior nodes only.

Comparison of Imaginary Part of Transfer Function, for Different Upwind Coefficients

Transfer functions are plotted for interior nodes only.

T.F

. (I

mag

inar

y)T

.F.

(Im

agin

ary)

Benefits of upwind filter

Problems of higher order central filters have been diagnosed as due to numerical instability near the boundary and excessive dissipation. These can be rectified by the upwind filter.

The upwind filter allows one to add controlled amount of dissipation in the interior of the domain. Absolute control over the imaginary part of the TF allows one to mimic the hyper-viscosity / SGS model used for LES.

Comparison of Numerical Amplification Factor Contours, for Different Upwind

Coefficients

Scaled numerical amplification contours for the solution of 1D wave equation, when the spatial discretization is carried out by OUCS3 and time discretization is obtained by RK4.

Scaled numerical group velocity contours for the solution of 1D wave equation, when the spatial discretization is carried out by OUCS3 and time discretization is obtained by RK4.

Comparison of Scaled Numerical Group Velocity Contours, With and Without

Upwind Filter

Experimental flow visualization picture (Thiria et. al. (2006))

Unfiltered solution

Filtered solution6th order azimuthal filter for30 lines, = 0.496th order radial filter forcomplete domain , = 0.45η = 0.001

Re = 150, A =2 ,ff/f0 = 1.5

t = 99.69

t = 100.99

t = 102.29

t = 99.69

t = 100.99

t = 102.29

Flow Direction

t = 1.92

Experimental visualization

8th order azimuthal filter = 0.48

6th order azimuthal filter = 0.48

6th order azimuthal filter for30 lines, = 0.485

6th order azimuthal filter for60 lines, = 0.485

5th order upwind wall-normal filter, η = 0.001

5th order upwind wall-normal filter, η = 0.001

Comparison of Flow Field Past NACA-0015 Airfoil

Flow Direction

t = 2.42

Experimental visualization8th order azimuthal filter = 0.48

6th order azimuthal filter = 0.48

6th order azimuthal filter for30 lines, = 0.485

6th order azimuthal filter for60 lines, = 0.485

5th order upwind wall-normal filter, η = 0.001

5th order upwind wall-normal filter, η = 0.001

Comparison of Flow Field Past NACA-0015 Airfoil

Recommended Filtering Strategy

Optimal filter is a combination of azimuthal central filter applied close to the wall with a non-periodic fifth order upwinded filter for the full domain.

This has similarity with DES (Barone & Roy (2006), Nishino et. al (2008) and Tucker (2003)), but does not require solving different equations for different parts of the domain.

One does not require turbulence or SGS models.

Hence this process is also computationally efficient in terms of computational efforts and cost.

Conclusions

1) Non-periodic filters cause numerical instability near inflow and excessive damping near outflow.

2) This problem can be removed by using a new upwind composite filter which even allows one to add controlled amount of dissipation in the interior of the domain.

3) Upwind filter has a better dispersion properties as compared to conventional symmetric filters.

4) This approach of using upwind filter does not require using different equations in the different parts of the domain. Also one does not need any turbulence or SGS models resulting in a fewer and faster computations.

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