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December, 2007VKI Lecture1 Boundary conditions

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December, 2007VKI Lecture2 BC essential for thermo-acoustics u’=0 p’=0 Acoustic analysis of a Turbomeca combustor including the swirler, the casing and the combustion chamber C. Sensiau (CERFACS/UM2) – AVSP code

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December, 2007VKI Lecture3 BC essential for thermo-acoustics C. Martin (CERFACS) – AVBP code

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December, 2007VKI Lecture4 Numerical test 1D convection equation (D=0) Initial and boundary conditions: Zero order extrapolation

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December, 2007VKI Lecture5 Numerical test t=3 t=6 t=9 t=12 t=15 t=18 t=21 t=24 t=27

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December, 2007VKI Lecture6 Basic Equations Primitive form: Simpler for analytical work Not included in wave decomposition

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December, 2007VKI Lecture7 Decomposition in waves in 1D A can be diagonalized: A = L -1 L - 1D Eqs: - Introducing the characteristic variables: - Multiplying the state Eq. by L:

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December, 2007VKI Lecture8 Remarks W i with positive (resp. negative) speed of propagation may enter or leave the domain, depending on the boundary in 3D, the matrices A, B and C can be diagonalized BUT they have different eigenvectors, meaning that the definition of the characteristic variables is not unique. M<1 u U + c U - c u U + c U - c

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December, 2007VKI Lecture9 Decomposition in waves: 3D Define a local orthonormal basiswith the inward vector normal to the boundary Introduce the normal matrix : Define the characteristic variables by:

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December, 2007VKI Lecture10 Which wave is doing what ? WAVE SPEEDINLET (u n >0) OUTLET (u n <0) W n 1 entropy unun inout W n 2 shear unun inout W n 3 shear unun inout W n 4 acoustic u n + cin W n 5 acoustic u n - cout

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December, 2007VKI Lecture11 General implementation Compute the predicted variation of V as given by the scheme of integration with all physical terms without boundary conditions. Note this predicted variation. Compute the corrected variation of the solution during the iteration as: Assess the corrected ingoing wave(s) depending on the physical condition at the boundary. Note its (their) contribution. Estimate the ingoing wave(s) and remove its (their) contribution(s). Notethe remaining variation.

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December, 2007VKI Lecture12 Pressure imposed outlet Compute the predicted value of P, viz. P P, and decompose it into waves: W n 4 is entering the domain; the contribution of the outgoing wave reads: The corrected value of W n 4 is computed through the relation: OK ! Desired pressure variation at the boundary The final (corrected) update of P is then:

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December, 2007VKI Lecture13 Defining waves: non-reflecting BC Very simple in principle: W n 4 =0 « Normal derivative » approach: « Full residual » approach: No theory to guide our choice … Numerical tests required

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December, 2007VKI Lecture14 1D entropy wave Same result with both the “normal derivative” and the “full residual” approaches

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December, 2007VKI Lecture15 2D test case A simple case: 2D inviscid shear layer with zero velocity and constant pressure at t=0 Full residual Normal residual

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December, 2007VKI Lecture16 Outlet with relaxation on P Start from Cut the link between ingoing and outgoing waves to make the condition non-reflecting Set to relax the pressure at the boundary towards the target value P t To avoid over-relaxation, P t should be less than unity. P t = 0 means ‘perfectly non-reflecting’ (ill posed)

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December, 2007VKI Lecture17 Inlet with relaxation on velocity and Temperature Cut the link between ingoing and outgoing waves Set to drive V B towards V t Use either the normal or the full residual approach to compute the waves and correct the ingoing ones via:

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December, 2007VKI Lecture18 Integral boundary condition in some situations, the target value is not known pointwise. E.g.: the outlet pressure of a swirled flow use the relaxation BC framework rely on integral values to generate the relaxation term to avoid disturbing the natural solution at the boundary

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December, 2007VKI Lecture19 Integral boundary condition periodic pulsated channel flow (laminar) U(y,t) / U bulk Integral BCs to impose the flow rate ?

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December, 2007VKI Lecture20 Everything is in the details Lodato, Domingo and Vervish – CORIA Rouen

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