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On The Dynamics of Mercury Target Delivery and Dump By Foluso Ladeinde Stony Brook University Stony Brook, New York Muon Collider Design Workshop December 1-3, 2009; at BNL

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Outline A Mechanical/Aerospace Engineer’s viewpoints on delivery, jet exhaust, and dump of Hg target –Whither Moody chart (friction factor) or system curve? - A different kind of pipe analysis –Whither the familiar decay laws for jet flows? – A different kind of jet analysis –Relevant analytical work on jet exhaust –CFD Analysis Scope Governing Equations The internal flow The jet exhaust The Dump The integrated model pipe-jet flow –Hybrid CFD-CSM-CFA Analysis

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The internal flow problem Pump/System Curve Moody Chart Whither Moody Chart (friction factor) and system curve?

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Whither the familiar jet decay laws?

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Analysis Options Data from physical experiments Exact, closed-form analysis Approximate methods –Computational fluid dynamics (fluid flow problems) –Computational solid mechanics (solid mechanics problems) –Computational material science (material science problems) –Computational dynamics analysis (dynamics problems) –Etc… Some advantages of numerical methods Focus on computational fluid dynamics (CFD)e Procedures for CFD analysis

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CFD-Governing Equations

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Turbulence Modeling Options in CFD RANS –Spalart-Allmaras –k- (Launder- Sharma) –k- (Abid model) –High Re No. k- –k- (Menter ’ s SST model) LES –Smagorinsky model –Dynamic SGS model –Implicit LES (Filtering) DES –Based on Spalart- Allmaras PRNS –Based on Abid k- –Based on high Re No. k- RANS/LES Hybrid

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CFD - Numerical Approach Spatial Discretization MUSCL (2 nd - order) WENO (5 th -order) COMPACT (6 th - order) Time Marching 2 nd order Beam-Warming 4 th order Runge-Kutta TVD Runge-Kutta

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CFD - MUSCL LR

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Back to physical space, Numerical flux Lax-Friedrichs Flux-splitting CFD - WENO WENO Reconstruction Transform to characteristic form:

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CFD - WENO… WENO Evaluation: Smoothness Indicator Robustness Factor Modifications Made:

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Filter high-frequency noise: COMPACT 6 th order theoretical accuracy:

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The internal flow problem DELIVERY SYSTEM CFD ANALYSIS – SBU (AEROFLO)

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The internal flow problem DELIVERY SYSTEM CFD ANALYSIS – SBU (AEROFLO)

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The internal flow problem DELIVERY SYSTEM CFD ANALYSIS – SBU (AEROFLO)

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The internal flow problem DELIVERY SYSTEM CFD ANALYSIS – SBU (AEROFLO)

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The internal flow problem DELIVERY SYSTEM CFD ANALYSIS – SBU (AEROFLO)

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The internal flow problem DELIVERY SYSTEM CFD ANALYSIS – SBU (AEROFLO)

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The internal flow problem DELIVERY SYSTEM CFD ANALYSIS – SBU (AEROFLO)

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Jet Flow –Relevant analytical work on jet exhaust in a magnetic field Osima et al. (1987) Field: 1/2 (1-tanh(z-15)/6.2) – inverted “S” Determination of the shape of the free surface Stuart number, Weber number, and ε m =a/L m determine shape of jet Round, elliptical, lobe-shaped Gallardo et al Field: Gaussian and other distributions (in z-) Changes in jet cross section and velocity assumed small! If jet enters field close to the axis, induced forces are compressive and retarding. Hydrostatic pressure and jet diameter increases, then re-accelerates and elongates. Hydrostatic pressure becomes negative and cavitation occurs as jet leaves field

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Instability wave Ansatz Circular nozzle –Parallel –Weakly non-parallel –Leads to ODE eigenvalue problem (in radial coordinate) –Efficient solution by shooting method Chevron nozzle –Parallel –Weakly non-parallel Leads to PDE eigenvalue problem in r,

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Pipe/Jet, RANS/LESHybrid Pipe/Jet RANS/LES,

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CFD – Jet Exhaust JET CFD ANALYSIS – TTC (AEROFLO) No MHD, energy input

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CFD – Jet Exhaust JET CFD ANALYSIS – TTC (AEROFLO) No MHD, energy input

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CFD – Turbulence Modeling LES / RANS

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CFD- Level-Set Equation (Peters) Turbulent Flame Speed Flame Curvature - Distance to the Flame Surface (Peters, Pitsch) Numerical Approach Spatial Discretization: ENO (up to 6 th order) Time Marching: TVD Runge-Kutta (2 nd, 3 rd order) Free Surface Calculation

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- pseudo-time is preserved numerical approach Re-initialization Procedure G-transport equation: Valid only at the flame surface Does not preserve the distance Sussman, Smereka, & Osher (1994) Russo & Smereka (2000) Sussman & Fatemi (1999) (for narrow-band method)

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Re-initialization Procedure Curvilinear Coordinates Central Finite Differences Upwinding based on W i

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CFD-Level set/VOF LEVEL SET METHOD – UCLA (HIMAG) By Others

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CFD-Level Set LEVEL SET – TTC (AEROFLO)

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DUMP: CFD – LEVEL SET LEVEL SET – RUTHERFORD APPLETON LAB(CFX) By Others – Tristan Davenne

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Dynamic Analysis – Dump material DYNAMIC ANALYSIS – RUTHERFORD (AUTODYNAMICS) By Others – Tristan Davenne

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Concluding Remarks Discussed delivery, jet exhaust, and dump of Hg target, from a mechanical/aerospace engineer viewpoint –No Moody chart, friction factor data, or system curves for operating point determination –A different kind of jet exhaust! –Couple of relevant analytical work on jet exhaust –CFD holds promise for the hybrid analysis –Level set (VOF) useful in determining free surface; thermal energy not built into procedures –Hybrid CFD-CSM-CFA Analysis More theoretical analysis needed for the internal flow and jet flow; extend with numerical procedures Dump analysis will most likely be numerical

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The Internal flow problem Mesh FileBlock #Grid SizeDescription pipe_01.grdBlock 1Pipe mesh in concentric cylinder form pipe_02.grdBlock 2Overset mesh to cover the centerline of pipe DELIVERY SYSTEM CFD ANALYSIS – SBU (AEROFLO)

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