Cartesian Schemes Combined with a Cut-Cell Method, Evaluated with Richardson Extrapolation
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Transcript of Cartesian Schemes Combined with a Cut-Cell Method, Evaluated with Richardson Extrapolation
Cartesian Schemes Combined with a Cut-Cell Method,
Evaluated with Richardson Extrapolation
D.N. Vedder
Prof. Dr. Ir. P. WesselingDr. Ir. C.Vuik
Prof. W. Shyy
Overview• Computational AeroAcoustics• Spatial discretization• Time integration• Cut-Cell method• Testcase
• Richardson extrapolation• Interpolation• Results• Conclusions
Computational AeroAcousticsAcoustics
• Sound modelled as an inviscid fluid phenomena Euler equations
• Acoustic waves are small disturbances Linearized Euler equations:
Computational AeroAcousticsDispersion relation
• A relation between angular frequency and wavenumber.
• Easily determined by Fourier transforms
Spatial discretization OPC
• Optimized-Prefactored-Compact scheme
1. Compact scheme
Fourier transforms and Taylor series
xj-2 xj-1 xj xj+1 xj+2
Spatial discretization OPC
• Taylor series
Fourth order gives two equations,this leaves one free parameter.
Spatial discretization OPC
• Fourier transformsTheorems:
Spatial discretization OPC
Spatial discretization OPC
Optimization over free parameter:
Spatial discretization OPC
2. Prefactored compact scheme
Determined by
Spatial discretization OPC
3. Equivalent with compact scheme
Advantages:1. Tridiagonal system two bidiagonal systems (upper and lower triangular)2. Stencil needs less points
Spatial discretization OPC
• Dispersive properties:
Time Integration LDDRK
• Low-Dissipation-and-Dispersion Runge-Kutta scheme
Time Integration LDDRK
• Taylor series• Fourier transforms• Optimization
• Alternating schemes
Time Integration LDDRK
Dissipative and dispersive properties:
Cut-Cell Method
• Cartesian grid• Boundary implementation
• Cut-cell method:– Cut cells can be merged– Cut cells can be independent
Cut-Cell Method
• fn and fw with boundary
stencils
• fint with boundary condition
• fsw and fe with interpolation polynomials which preserve 4th order of accuracy. (Using neighboring points)
fn
fw
fsw fint
fe
TestcaseReflection on a solid wall• Linearized Euler
equations
• Outflow boundaryconditions
• 6/4 OPC and 4-6-LDDRK
ResultsPressure contours
The derived order of accuracy is 4.
What is the order of accuracy in practice?
What is the impact of the cut-cell method?
Richardson extrapolation
Determining the order of accuracy
Assumption:
Richardson extrapolation
Three numerical solutions needed
Pointwise approach interpolation to a common grid needed
InterpolationInterpolation polynomial:Fifth degree in x and y 36 points
1. Lagrange interpolation in interior– 6x6 squares
2. Matrix interpolation near wall– Row Scaling– Shifting interpolation procedure– Using wall condition
6th order interpolation method, tested by analytical testcase
ResultsSolution at t = 4.2 Order of accuracy at t = 4.2
Results (cont)Impact of boundary condition and filter
• Boundary condition
• Filter for removing high frequency noise
Results (cont)Order of accuracy
t = 4.2 t = 8.4
Results (cont)Impact of outflow condition
• Outflow boundary condition
• Replace by solid wall
Results (cont)Impact of cut-cell method
Order of accuracy
t = 8.4 t = 12.6
Solid wall
Results (cont)Impact of cut-cell method
• Interpolation method used for and
• Tested by analytical testcase
• Results obtained with three norms– Order of accuracy about 0!!
fn
fw
fsw fint
fe
fsw fe
Results (cont)Richardson extrapolation
Results (cont)Richardson extrapolation
Conclusions• Interpolation to common grid
– 6th order to preserve accuracy of numerical solution
• Impact of discontinuities and filter– Negative impact on order of accuracy
• Impact of outflow boundary conditions– Can handle waves from only one direction
• Impact of cut-cell method– Lower order of accuracy due to interpolation
• Richardson extrapolation– Only for “smooth” problems
Questions?