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

Real part Imaginary part Transfer Function Variation At Different Nodes for the Sixth Order Non-Periodic Filter Transfer function overshoot for near boundary nodes Numerical instability at nodes 2 and 3 Excessive dissipation at nodes N-1 and N-2 Imaginary parts affect dispersion properties T.F. (Real) T.F. ( Imaginary )

Real part of the Transfer Function for the Composite Filter ** At nodes 2 and N-1 : Second order filter stencil 3 and N-2 : Fourth order filter stencil 4 to N-3 : Sixth order filter stencil ** LOC approach – Chu & Fan (JCP 1998), Gaitonde & Visbal (AIAA 2000)

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 = 0.2π Nc = 0.1 C = 0.1

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 Filter, 6 th Order Filter

Flow Around a Cylinder Performing Rotary Oscillation.

Numerical Stabilization Property of the Filters Recommended central filters: 1) 4 th order filter with  value very close to 0.5 2) 2 nd order filter apply infrequently

Effect of Frequency of Filtering on the Computed Solution 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).

Experimental flow visualization picture (Thiria et. al. (2006)) Unfiltered solution Filtered solution 6th order azimuthal filter for 30 lines,  = th order radial filter for complete domain,  = 0.45 Re = 150, A =2,f f /f 0 = 1.5

Transfer Function Variation At Different Nodes for the Given Filter Coefficients T.F. (Real) Real Part Imaginary Part T.F. (Imaginary)

T.F. ( Real ) 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. ( Imaginary )

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 solution 6th order azimuthal filter for 30 lines,  = th order radial filter for complete domain,  = 0.45 η = Re = 150, A =2,f f /f 0 = 1.5

Flow Direction t = 1.92 Experimental visualization 8 th order azimuthal filter  = th order azimuthal filter  = th order azimuthal filter for 30 lines,  = th order azimuthal filter for 60 lines,  = th order upwind wall-normal filter, η = th order upwind wall-normal filter, η = Comparison of Flow Field Past NACA-0015 Airfoil

Flow Direction t = 2.42 Experimental visualization 8 th order azimuthal filter  = th order azimuthal filter  = th order azimuthal filter for 30 lines,  = th order azimuthal filter for 60 lines,  = th order upwind wall-normal filter, η = th order upwind wall-normal filter, η = 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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