[PPT] - Basic hydrodynamics and Introduction to FLASH Kuo-Chuan Pan ( ) PowerPoint Presentation

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Numerical Astrophysics Summer School: Computational Astrophysical Fluid Dynamics, 2019.09.04, IoA, NTHU

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Basic hydrodynamics and Introduction to FLASH

Kuo-Chuan Pan (潘國全) Institute of Astronomy, NTHU

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Outline:

Astrophysical fluids
Basic hydrodynamic equations
Introduction to the FLASH code

NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

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Why we need simulations in Astronomy?

We receive and record, but we do not

experiment or perturb (too distant)

Astrophysical systems involve extreme

conditions that cannot be replicated in lab (mostly)

Timescales are >> human lifetime (mostly)

NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

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Telescope Astrophysical object Detectors Data reduction / Calibration

Observation

DATA

Observation and Computation Analogies

Mathematical Model Supercomputer DATA Data analysis / Visualization

PN

cool

Codes

Simulation 4

NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

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Frontier Astronomy Research

Observational Astronomy Theoretical Astrophysics Computational Astrophysics Hypothesis Observations Formulation Realization Validation Postdiction, Prediction Michael Norman (1997)

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NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

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Astrophysical Problems

Planets (~ Earth radius) Stars/Binaries (~0.1-1000 AU) Planetary disk (~100-1000 AU) Interstellar medium (~10-100 pc) Galaxy (~50 kpc) Galaxy clusters (~1 Mpc) Large scale structure (~1 Gpc)

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NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

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Astrophysical Simulations

2012 ApJ, 750, 151 2012 ApJ, 746, 74 2009 MNRAS 398, 1082 2005 ApJS 160, 28 2013MNRAS

2018 ApJ

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NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

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Fluid Approximation

To solve the gas dynamics equations,
ne could use N-body method if we

know the force between two particles.

However, the number of particles is

usually huge (NA=1023). It is impossible to use direct N-body method to study the gas dynamics. -> Fluid

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NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

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Fluid Approximation (continue)

L >> λmfp

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T >> τcoll

<latexit sha1_base64="JzZ5jwemKs9qjTxWXhqtJoF0+9E=">AB/XicbVDLSsNAFL3xWesrPnZuBovgqiRV0FUpunFZoS9oQphMp+3QySTMTIQair/ixoUibv0Pd/6N0zYLbT1w4XDOvTP3njDhTGnH+bZWVtfWNzYLW8Xtnd29fvgsKXiVBLaJDGPZSfEinImaFMzWknkRHIaftcHQ79dsPVCoWi4YeJ9SP8ECwPiNYGymwjxuoWkWexmQeTJC5kU+CeySU3ZmQMvEzUkJctQD+8vrxSNqNCEY6W6rpNoP8NSM8LpOiliaYjPCAdg0VOKLKz2bT9CZUXqoH0tTQqOZ+nsiw5FS4yg0nRHWQ7XoTcX/vG6q+9d+xkSairI/KN+ypGO0TQK1GOSEs3HhmAimdkVkSGWmGgTWNGE4C6evExalbJ7Ua7cX5ZqN3kcBTiBUzgHF6gBndQhyYQeIRneIU368l6sd6tj3nripXPHMEfWJ8/XH6UiQ=</latexit>

, Mean free path , Collision time scale

Exercise: Consider the air in this room

ρ ∼ 10−3 g/cm3, µ ∼ 28mp, T ∼ 300 K

<latexit sha1_base64="m2XjDjqjtSgx1hefcFBaPuJUtw=">ACKHicbZDLSsNAFIYnXmu9RV26GWwEF1qTdmF3Ft0Ibir0Bk1bJtNpO3QmCTMToYT6Nm58FTcinTrkzhps9DWAwM/38OZ87vhYxKZdtTY2V1bX1jM7OV3d7Z3ds3Dw7rMogEJjUcsEA0PSQJoz6pKaoYaYaCIO4x0vBGt4nfeCRC0sCvqnFI2hwNfNqnGCmNua15YphAF1JOXTsTnxRnFhwcIm51Sla59ByeTQ3CyXIu2GCqnNQtG3r6b5r5uy8PSu4LJxU5EBala757vYCHiK8yQlC3HDlU7RkJRzMgk60aShAiP0IC0tPQRJ7Idzw6dwFNerAfCP18BWf090SMuJRj7ulOjtRQLnoJ/M9rRapfasfUDyNFfDxf1I8YVAFMUoM9KghWbKwFwoLqv0I8RAJhpbPN6hCcxZOXRb2Qd4r5wkMhV75J48iAY3ACzoADrkAZ3IEKqAEMnsEr+ACfxovxZnwZ03nripHOHIE/ZXz/ABjIoRE=</latexit>

Cross section σ ∼ 3 × 10−15 cm2

<latexit sha1_base64="mambz7N7ThaAgYcCbQ+8zgdybKI=">ACHXicbVC7TsMwFHXKq5RXgJHFokVioUpSEIwVXRiLRB9Sk1aO67ZW7SyHaQqKh/Cwq+wMIAQAwvib3DaDNByJMvH59yr63v8iFGpLOvbyK2srq1v5DcLW9s7u3vm/kFThrHApIFDFoq2jyRhNCANRUj7UgQxH1GWv64lvqteyIkDYM7NYmIx9EwoAOKkdJSzyviVBKAlO37DkSjrkCOqLwp0FeVEQtvqJmf2xbT0gHmp65R6ZtEqWzPAZWJnpAgy1Hvmp9sPcxJoDBDUnZsK1JegoSimJFpwY0liRAeoyHpaBogPdVLZtN4YlW+nAQCn0CBWfq74EcSkn3NeVHKmRXPRS8T+vE6vBlZfQIoVCfB80CBmUIUwjQr2qdCxsIkmCAuq/wrxCAmElQ60oEOwF1deJk2nbFfKzq1TrF5nceTBETgGp8AGl6AKbkAdNAGj+AZvI348l4Md6Nj3lpzsh6DsEfGF8/+if7w=</latexit>

RMS speed of nitrogen v ∼ q

8kT πµ ∼ 5 × 104 cm/s

<latexit sha1_base64="Qd4lhLzqkC7HcCqFoB0Y2EY7f9U=">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</latexit>

τcoll ∼ (nσv)−1 ∼ 3 × 10−10 sec

<latexit sha1_base64="aKTG/YG1E/O4qCrYkfdjElItw=">ACLHicbVDLSgMxFM3Ud32NunQTrIuLDPtQpdiNy4r2Fbo1J0xqaZIbkTqEM9X/c+CuCuFDErd9hps5CWw+EHM65Nzf3hLHgBjzv3SksLC4tr6yuFdc3Nre23Z3dpokSTVmDRiLStyExTHDFGsBsNtYMyJDwVrhsJb5rRHThkfqBsYx60gyULzPKQErd3aYQAk6aBlti+JiY4MFziY5XdA0nw6OQuPfVzuYoD4JIZ7HuZ6k0OHwyjXbfklb0p8Dzxc1JCOepd9yXoRTSRTAEVxJi278XQSYkGTgWbFIPEsJjQIRmwtqWK2JGdLrsB9ZpYf7kbZHAZ6qvztSIo0Zy9BWSgL3ZtbLxP+8dgL9807KVZwAU/RnUD8RGCKcJYd7XDMKYmwJoZrbv2J6TzShYPMt2hD82ZXnSbNS9qvlynWldHGZx7GK9tEBOkY+OkMX6ArVUQNR9Iie0Rt6d56cV+fD+fwpLTh5zx76A+frG4KzpeQ=</latexit>

λmfp ∼ (nσ)−1 ∼ 3 × 10−5 cm

<latexit sha1_base64="cXule+3v6zlYz0qLy32x8msujq8=">ACKnicbVDLTgIxFO3gC/E16tJNI5jgQjIDMbpE3bjERB4Jg6RTOtDQdiZtx4RM8Hfc+CtuWGiIWz/EArNQ8CZNT845N/fe40eMKu04Uyuztr6xuZXdzu3s7u0f2IdHDRXGEpM6DlkoWz5ShFB6pqRlqRJIj7jDT94d1Mbz4TqWgoHvUoIh2O+oIGFCNtqK59U/CYcfdQN/EkhzyIxtBTlMOimP19js6fkgs3JSvQ05QTBV3HsJfjwgvmXTvlJx5wVXgpiAP0qp17YnXC3HMidCYIaXarhPpToKkpiRc6LFYkQHqI+aRsokBnYSeanjuGZYXowCKV5QsM5+7sjQVypEfeNkyM9UMvajPxPa8c6uO4kVESxJgIvBgUxgzqEs9xgj0qCNRsZgLCkZleIB0girE26OROCu3zyKmiUS26lVH4o56u3aRxZcAJOQRG4ApUwT2ogTrA4BW8gw/wab1ZE2tqfS2sGSvtOQZ/yvr+AVGepVM=</latexit>

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NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

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Fluid Approximation (continue)

L >> λmfp

<latexit sha1_base64="A+i98ATuwWo9K2AUbMFdinZkek8=">AB/3icbVBPS8MwHE3nvzn/VQUvXoJD8DTaKehpDL148DBbcJaSpqmW1iSliQVRt3Br+LFgyJe/Rre/DZmWw+6+SDweO/3+P3ywpRpR3n2yotLa+srpXKxubW9s79u5eRyWZxKSNE5bI+xApwqgbU01I/epJIiHjHTD4dXE7z4QqWgi7vQoJT5HfUFjipE2UmAf3MBGA3rMJCIU5J7kMfpOLCrTs2ZAi4StyBVUKAV2F9elOCME6ExQ0r1XCfVfo6kpiRcXLFEkRHqI+6RkqECfKz6f3j+GxUSIYJ9I8oeFU/Z3IEVdqxEMzyZEeqHlvIv7n9TIdX/g5FWmicCzRXHGoE7gpAwYUmwZiNDEJbU3ArxAEmEtamsYkpw57+8SDr1mntaq9+eVZuXR1lcAiOwAlwTlogmvQAm2AwSN4Bq/gzXqyXqx362M2WrKzD74A+vzB7JElU=</latexit>

T >> τcoll

<latexit sha1_base64="JzZ5jwemKs9qjTxWXhqtJoF0+9E=">AB/XicbVDLSsNAFL3xWesrPnZuBovgqiRV0FUpunFZoS9oQphMp+3QySTMTIQair/ixoUibv0Pd/6N0zYLbT1w4XDOvTP3njDhTGnH+bZWVtfWNzYLW8Xtnd29fvgsKXiVBLaJDGPZSfEinImaFMzWknkRHIaftcHQ79dsPVCoWi4YeJ9SP8ECwPiNYGymwjxuoWkWexmQeTJC5kU+CeySU3ZmQMvEzUkJctQD+8vrxSNqNCEY6W6rpNoP8NSM8LpOiliaYjPCAdg0VOKLKz2bT9CZUXqoH0tTQqOZ+nsiw5FS4yg0nRHWQ7XoTcX/vG6q+9d+xkSairI/KN+ypGO0TQK1GOSEs3HhmAimdkVkSGWmGgTWNGE4C6evExalbJ7Ua7cX5ZqN3kcBTiBUzgHF6gBndQhyYQeIRneIU368l6sd6tj3nripXPHMEfWJ8/XH6UiQ=</latexit>

, Mean free path , Collision time scale

Exercise: In ISM

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NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

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Cross section σ ∼? cm2

<latexit sha1_base64="4Dq1RNdxAa+Li7WRChbCOEJco5o=">ACDXicbVC7TsMwFHXKq5RXgJHFokFiqpIywEZF8Yi0YfUhspxndaq7US2g1RF5QNY+BUWBhBiZWfjb3DaDNByJMvH59yr63uCmFGlXfbKqysrq1vFDdLW9s7u3v2/kFLRYnEpIkjFslOgBRhVJCmpqRTiwJ4gEj7WBcz/z2PZGKRuJWT2LiczQUNKQYaSP1bacuI6WgIjh7Q6en6JAjaC4OL50HzJ27qtO3y27FnQEuEy8nZCj0be/eoMIJ5wIjRlSqu5sfZTJDXFjExLvUSRGOExGpKuoQJxovx0ts0UnhlAMNImiM0nKm/O1LElZrwFRypEdq0cvE/7xuosMLP6UiTjQReD4oTBjUEcyigQMqTQxsYgjCkpq/QjxCEmFtAiyZELzFlZdJq1rxzirVm2q5dpXHUQRH4BicAg+cgxq4Bg3QBg8gmfwCt6sJ+vFerc+5qUFK+85BH9gf4ALseaUw=</latexit>

number density n ∼? cm−3

<latexit sha1_base64="hS7SI3+hvDO379QV4+Z9SH3gHo=">ACDHicbVC5TgMxFPRyhnAFKGkskg0RLtJAR0RNJRBIoeULJHXeUms2N6V7UWKVqGn4VdoKECIlg+g429wjgISRrI0mpmn5zdhzJk2nvftLC2vrK6tZzaym1vbO7u5vf2ajhJFoUojHqlGSDRwJqFqmOHQiBUQEXKoh4OrsV+/B6VZJG/NMIZAkJ5kXUaJsVI7l5eJCEHhDkjNzBC7Erc0E/jCfaDCvUtPSyPXpryCNwFeJP6M5NEMlXbuq9WJaCJAGsqJ1k3fi02QEmUY5TDKthINMaED0oOmpZI0E6OWaEj63Swd1I2ScNnqi/J1IitB6K0CYFMX09743F/7xmYrnQcpknBiQdLqom3BsIjxuBneYAmr40BJCFbN/xbRPFKHG9pe1JfjzJy+SWrHglwrFm2K+fDmrI4MO0RE6QT46Q2V0jSqoih6RM/oFb05T86L8+58TKNLzmzmAP2B8/kDhXuZ+g=</latexit>

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Fluid Approximation (continue)

L >> λmfp

<latexit sha1_base64="A+i98ATuwWo9K2AUbMFdinZkek8=">AB/3icbVBPS8MwHE3nvzn/VQUvXoJD8DTaKehpDL148DBbcJaSpqmW1iSliQVRt3Br+LFgyJe/Rre/DZmWw+6+SDweO/3+P3ywpRpR3n2yotLa+srpXKxubW9s79u5eRyWZxKSNE5bI+xApwqgbU01I/epJIiHjHTD4dXE7z4QqWgi7vQoJT5HfUFjipE2UmAf3MBGA3rMJCIU5J7kMfpOLCrTs2ZAi4StyBVUKAV2F9elOCME6ExQ0r1XCfVfo6kpiRcXLFEkRHqI+6RkqECfKz6f3j+GxUSIYJ9I8oeFU/Z3IEVdqxEMzyZEeqHlvIv7n9TIdX/g5FWmicCzRXHGoE7gpAwYUmwZiNDEJbU3ArxAEmEtamsYkpw57+8SDr1mntaq9+eVZuXR1lcAiOwAlwTlogmvQAm2AwSN4Bq/gzXqyXqx362M2WrKzD74A+vzB7JElU=</latexit>

T >> τcoll

<latexit sha1_base64="JzZ5jwemKs9qjTxWXhqtJoF0+9E=">AB/XicbVDLSsNAFL3xWesrPnZuBovgqiRV0FUpunFZoS9oQphMp+3QySTMTIQair/ixoUibv0Pd/6N0zYLbT1w4XDOvTP3njDhTGnH+bZWVtfWNzYLW8Xtnd29fvgsKXiVBLaJDGPZSfEinImaFMzWknkRHIaftcHQ79dsPVCoWi4YeJ9SP8ECwPiNYGymwjxuoWkWexmQeTJC5kU+CeySU3ZmQMvEzUkJctQD+8vrxSNqNCEY6W6rpNoP8NSM8LpOiliaYjPCAdg0VOKLKz2bT9CZUXqoH0tTQqOZ+nsiw5FS4yg0nRHWQ7XoTcX/vG6q+9d+xkSairI/KN+ypGO0TQK1GOSEs3HhmAimdkVkSGWmGgTWNGE4C6evExalbJ7Ua7cX5ZqN3kcBTiBUzgHF6gBndQhyYQeIRneIU368l6sd6tj3nripXPHMEfWJ8/XH6UiQ=</latexit>

, Mean free path , Collision time scale

Exercise: In ISM

Cross section σ ∼ 3 × 10−15 cm2

<latexit sha1_base64="mambz7N7ThaAgYcCbQ+8zgdybKI=">ACHXicbVC7TsMwFHXKq5RXgJHFokVioUpSEIwVXRiLRB9Sk1aO67ZW7SyHaQqKh/Cwq+wMIAQAwvib3DaDNByJMvH59yr63v8iFGpLOvbyK2srq1v5DcLW9s7u3vm/kFThrHApIFDFoq2jyRhNCANRUj7UgQxH1GWv64lvqteyIkDYM7NYmIx9EwoAOKkdJSzyviVBKAlO37DkSjrkCOqLwp0FeVEQtvqJmf2xbT0gHmp65R6ZtEqWzPAZWJnpAgy1Hvmp9sPcxJoDBDUnZsK1JegoSimJFpwY0liRAeoyHpaBogPdVLZtN4YlW+nAQCn0CBWfq74EcSkn3NeVHKmRXPRS8T+vE6vBlZfQIoVCfB80CBmUIUwjQr2qdCxsIkmCAuq/wrxCAmElQ60oEOwF1deJk2nbFfKzq1TrF5nceTBETgGp8AGl6AKbkAdNAGj+AZvI348l4Md6Nj3lpzsh6DsEfGF8/+if7w=</latexit>

number density n ∼ 1 cm−3

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λmfp ∼ (nσ)−1 ∼ 1015 cm

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τcoll ∼ (nσv)−1 ∼ 109 sec ∼ 40 yrs

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Sun

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Star forming region (S106)

Image credit: NASA

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Cas A Supernova Remnant

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Image credit: NASA

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Bullet cluster

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Image credit: NASA

0.5 Mpc

SLIDE 16

Astrophysical Fluids

https://youtu.be/ICxC5ekWnUc

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Astrophysical fluids

Different with

Earth-bound fluid

Wide range of

physical Conditions

Sun Neutron star Earth ISM IGM Molecular cloud SgrA* 17

NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

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Astrophysical fluids

Sun Neutron star Earth ISM IGM Molecular cloud SgrA*

Magnetic fields are often

important

Compressibility is often

important

Occasionally self-gravitating

and relativistic

Non-ideal process are often

unimportant

Sometimes involves

interactions with non- thermal process and strong radiation fields

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SLIDE 19

Governing equations

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NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

19

The governing equations for ideal (inviscid)

hydrodynamics. The main physical ideas are

simple:

Mass is conserved (1 constraint)
Momentum is conserved (3 constraints)
Energy is conserved (1 constraint)

variables: ρ, v, and u

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Gas density Gas velocity Gas internal energy

ρ

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v

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u

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SLIDE 20

Coordinates

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NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

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Eulerian Lagrangian

SLIDE 21

Conservational laws

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NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

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The conservational laws can be written in what is called “conservation form”:

∂t(U) = r · F

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F

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U

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Density of a quantity Flux density for that quantity

Z

V

(r · F ) dV = I

S

F · ndS

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“Gauss’s Theorem" or "divergence theorem”

F

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F

<latexit sha1_base64="PfKtAJFRnjHMo8OwSjq5S1mVQxc=">AB9XicdVBLSwMxGMzWV62vqkcvwSJ4WrLtYru3oiAeK9gHtGvJZrNtaPZBklXK0v/hxYMiXv0v3vw3ZtsKjoQMsx8H5mMl3AmFUIfRmFldW19o7hZ2tre2d0r7x90ZJwKQtsk5rHoeVhSziLaVkx2ksExaHadebXOR+94KyeLoRk0T6oZ4FLGAEay0dDvwYu7Laiv7HI2LFeQ6TQaDrIhMm2E7HpVk5rjOGcIWiaowKWaA3L7wM/JmlI0U4lrJvoUS5GRaKEU5npUEqaYLJBI9oX9MIh1S62Tz1DJ5oxYdBLPSJFJyr3zcyHMo8mp4MsRrL314u/uX1UxU03IxFSapoRBYPBSmHKoZ5BdBnghLFp5pgIpjOCskYC0yULqkS/j6KfyfdKqmVTOr13aleb6sowiOwDE4BRaogya4Ai3QBgQI8ACewLNxbzwaL8brYrRgLHcOwQ8Yb5+MgpM2</latexit>

SLIDE 22

Continuity equation

∂ρ ∂t + r · (ρv) = 0

<latexit sha1_base64="dUBsrN4bCT5H6ytEHTBdCz2GjaE=">ACNHicbVBNSwMxEM36WetX1aOXYBEUoexWQS9C0YvgpYJVoVvKbDbBrPJkmQLZdkf5cUf4kUED4p49TeYrQU/6kDImzdvmJkXJxp47pPztT0zOzcfGmhvLi0vLJaWVu/0jJVhLaI5FLdBKApZ4K2DOc3iSKQhxweh3cnhb16wFVmklxaYJ7cTQEyxiBIylupVzP1JAMj8BZRhw7Ku+zL9Tk+M97AsIOGCfhNLgnUKB/UDyUA9j+5WzQb6Lj7Fb7laqbs0dBZ4E3hU0Tia3cqDH0qSxlQYwkHrtucmpMVswmnedlPNU2A3EKPti0UEFPdyUZH53jbMiGOpLJPGDxif3ZkEOtiQ6uMwfT131pB/ldrpyY6mRMJKmhgnwNilLrhcSFgzhkihLDhxYAUczuikfrIvG+lyY4P09eRJc1Wvefq1+cVBtnIztKFNtIV2kIcOUQOdoSZqIYLu0CN6Qa/OvfPsvDnvX9IpZ9yzgX6F8/EJBlKqUw=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

v

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Gas density Gas velocity

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NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

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Z

V

(r · F ) dV = I

S

F · ndS

<latexit sha1_base64="Ap73pE2/3UVwDNqw3paXvJmShI=">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</latexit>

“Gauss’s Theorem" or "divergence theorem”

F

<latexit sha1_base64="PfKtAJFRnjHMo8OwSjq5S1mVQxc=">AB9XicdVBLSwMxGMzWV62vqkcvwSJ4WrLtYru3oiAeK9gHtGvJZrNtaPZBklXK0v/hxYMiXv0v3vw3ZtsKjoQMsx8H5mMl3AmFUIfRmFldW19o7hZ2tre2d0r7x90ZJwKQtsk5rHoeVhSziLaVkx2ksExaHadebXOR+94KyeLoRk0T6oZ4FLGAEay0dDvwYu7Laiv7HI2LFeQ6TQaDrIhMm2E7HpVk5rjOGcIWiaowKWaA3L7wM/JmlI0U4lrJvoUS5GRaKEU5npUEqaYLJBI9oX9MIh1S62Tz1DJ5oxYdBLPSJFJyr3zcyHMo8mp4MsRrL314u/uX1UxU03IxFSapoRBYPBSmHKoZ5BdBnghLFp5pgIpjOCskYC0yULqkS/j6KfyfdKqmVTOr13aleb6sowiOwDE4BRaogya4Ai3QBgQI8ACewLNxbzwaL8brYrRgLHcOwQ8Yb5+MgpM2</latexit>

F

<latexit sha1_base64="PfKtAJFRnjHMo8OwSjq5S1mVQxc=">AB9XicdVBLSwMxGMzWV62vqkcvwSJ4WrLtYru3oiAeK9gHtGvJZrNtaPZBklXK0v/hxYMiXv0v3vw3ZtsKjoQMsx8H5mMl3AmFUIfRmFldW19o7hZ2tre2d0r7x90ZJwKQtsk5rHoeVhSziLaVkx2ksExaHadebXOR+94KyeLoRk0T6oZ4FLGAEay0dDvwYu7Laiv7HI2LFeQ6TQaDrIhMm2E7HpVk5rjOGcIWiaowKWaA3L7wM/JmlI0U4lrJvoUS5GRaKEU5npUEqaYLJBI9oX9MIh1S62Tz1DJ5oxYdBLPSJFJyr3zcyHMo8mp4MsRrL314u/uX1UxU03IxFSapoRBYPBSmHKoZ5BdBnghLFp5pgIpjOCskYC0yULqkS/j6KfyfdKqmVTOr13aleb6sowiOwDE4BRaogya4Ai3QBgQI8ACewLNxbzwaL8brYrRgLHcOwQ8Yb5+MgpM2</latexit>

SLIDE 23

Convective derivative

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NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

23

Consider a fluid element

v = dr dt ,

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a = dv dt ,

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a = dv(r, t) dt

<latexit sha1_base64="nF+KZMxGK3qzWe3Y3VgHFk14lJk=">ACJnicbVDLSgMxFM3UV62vqks3waJUkDLTLnRhoeDGZQX7gLaUTCbThmYeJHcKZivceNP+AFuXFRE3PkB/oJg+ljU1gMhJ+fcS+49di4AtP8NFJr6xubW+ntzM7u3v5B9vCoroJIUlajgQhk0yaKCe6zGnAQrBlKRjxbsIY9uJ34jSGTigf+A4xC1vFIz+cupwS01M2W23YgHDXy9BWTBJdx25WExs6iPkzyi095CRdJEjuQdLM5s2BOgVeJNSe5Sqn5/HNufle72XHbCWjkMR+oIEq1LDOETkwkcCpYkmlHioWEDkiPtT1icdUJ56umeAzrTjYDaQ+PuCputgRE09NRtSVHoG+WvYm4n9eKwL3uhNzP4yA+XT2kRsJDAGeZIYdLhkFMdKEUMn1rJj2iY4JdLIZHYK1vPIqRcLVqlQvNdp3KAZ0ugEnaI8stAVqA7VEU1RNEjekFj9GY8Ga/Gu/ExK0Z85j9AfG1y9lxqtK</latexit>

= ∂v ∂t + ∂v ∂r · ∂r ∂t

<latexit sha1_base64="mWY+S8dazQD2+wNf/rTdtycY9mg=">ACTXicfVHLSgMxFM3UR2t9jbp0EyKIJSZdqELhYIblxXsAzqlZDKZNpiZDEmUIb5NH/AjeDOf3DhxoWiYqYtaFvxQuBwzrk3uSduxKhUlvVk5JaWV1bzhbXi+sbm1ra5s9uUPBaYNDBnXLRdJAmjIWkoqhpR4KgwGWk5d5eZnprSISkPLxRo4h0A9QPqU8xUprqmR68gI4vE6cCAlFEYPD9AerFJ78p4sUOtjat4jZmb0zJVtsYF4E9BaVatX3eWQ913vmo+NxHAckVJghKTu2Falukg3EjKRFJ5YkQvgW9UlHwxAFRHaTcRopPNSMB30u9AkVHLO/OxIUSDkKXO0MkBrIeS0j/9I6sfLPugkNo1iRE8u8mO9IdZtNCjgmDFRhogLKh+K8QDpHNR+gOKOgR7fuVF0KyU7Wq5cq3TOAeTKoB9cACOgQ1OQ1cgTpoAzuwQt4A+/Gg/FqfBhfE2vOmPbsgZnK5b8BITm5aQ=</latexit>

= ∂v ∂t + (v · r)v

<latexit sha1_base64="tPIiEzXp2TvwAnPbJ5lxCy4GEv8=">ACHnicbVDLSgMxFM34tr6qLt0ERakIZaZFdKEguHFZwdZCZyh30kwbmskMSaZQhvkSN/6ICzcuFBFc6T/4C4JpK6itBwKHc+7l5hw/5kxp2363pqZnZufmFxZzS8srq2v59Y2aihJaJVEPJ1HxTlTNCqZprTeiwphD6n1373fOBf96hULBJXuh9TL4S2YAEjoI3UzB/iU+wGEkjqxiA1A4572Q/XGT7AhR52SvS2BXgc9jHvWZ+xy7aQ+BJ4nyTnbNy/e5z/6oNPOvbisiSUiFJhyUajh2rL10cIVwmuXcRNEYSBfatGogJAqLx3Gy/CuUVo4iKR5QuOh+nsjhVCpfuibyRB0R417A/E/r5Ho4NhLmYgTQUZHQoSkzrCg65wi0lKNO8bAkQy81dMOmDK0qbRnCnBGY8SWqlolMuli5NGydohAW0hbZRATnoCJ2hC1RBVUTQDbpHj+jJurUerGfrZTQ6ZX3vbKI/sN6+AExFpQc=</latexit>

≡ Dv Dt

<latexit sha1_base64="4tP6a28jLClu43w8FV/y0hljG8=">AB/nicbVC7SgNBFJ31GeNrVaxsBoNiFXaTQguLgCksI5gHZJcwO5lNhsw+nLkbCMuCv2JjoYitjT9h5z/4C4KTR6GJBy4czrmXe+/xYsEVWNansbS8srq2ntvIb25t7+yae/sNFSWSsjqNRCRbHlFM8JDVgYNgrVgyEniCNb3B1dhvDplUPApvYRQzNyC9kPucEtBSxzEDrtL+BA7viQ0rQ6ztApZxyxYRWsCvEjsGSlUyq371Prq9YxP5xuRJOAhUAFUaptWzG4KZHAqWBZ3kUiwkdkB5raxqSgCk3nZyf4ROtdLEfSV0h4In6eyIlgVKjwNOdAYG+mvfG4n9eOwH/wk15GCfAQjpd5CcCQ4THWeAul4yCGlCqOT6Vkz7ROcAOrG8DsGef3mRNEpFu1ws3eg0LtEUOXSEjtEZstE5qBrVEN1RFGKHtATejbujUfjxXidti4Zs5kD9AfG2w87+Jle</latexit>

Convective derivate

Rewrite continuity equation

∂tρ = r · (ρv)

<latexit sha1_base64="fsnR1CKXZ2insp0uvezpHVZ0/so=">ACH3icbVDLSgMxFM34rPVdamLoAh1YZmpoC4UBDcuFawKnVLuZFIbzCRDckcoQ7/BH3Djr3TjQhFx54e4N9O68HUg5HDOvdx7T5RKYdH372x8YnJqenSTHl2bn5hsbK0fGF1ZhvMC21uYrAcikUb6BAya9SwyGJL+Mbo4L/KWGyu0OsdeylsJXCvREQzQSe3KbpiCQGyjTQ0XU0P6TYNFUQSaMhijbQ6lMNIy9j2Evflt/2tdmXDr/lD0L8k+CIbR2t3YfVjcHfaryFsWZwhUyCdY2Az/FVl7MZpL3y2FmeQrsBq501EFCbetfHhfn246JaYdbdxTSIfq94cElvs5ioTwK797RXif14zw85+KxcqzZArNhrUySRFTYuwaCwMZyh7jgAzwu1KWRcMHSRl0Iwe+T/5KLei3YqdXPXBoHZIQSWSXrpEoCskeOyAk5JQ3CyD0ZkCfy7D14j96L9zoqHfO+elbID3jvn6y7pdg=</latexit>

= (v · r)ρ ρ(r · v)

<latexit sha1_base64="6NvTD8pX1r1nIWU6eEuMVEQ2ek=">ACGHicbZDLSgMxFIYz9VbrbdSlLoJFaBfWmbrQhULBjcsK9gKdUjJp2oZmkiHJFErpM3TlxldxI6KI2+58EPdmOl1o64GQn+8/h+T8fsio0o7zZaVWVtfWN9Kbma3tnd09e/+gqkQkMalgwYSs+0gRjmpaKoZqYeSoMBnpOb3b2O/NiBSUcEf9DAkzQB1Oe1QjLRBLfv8Bp7B3AB6uC09DjyGcp7sicMjq9cgub+IN+ys07BmRVcFu5cZEvHEy/3/Topt+yp1xY4CgjXmCGlGq4T6uYISU0xI+OMFykSItxHXdIwkqOAqOZotgYnhrSh0hzeEazujviREKlBoGvukMkO6pRS+G/3mNSHeumiPKw0gTjpOHOhGDWsA4JdimkmDNhkYgLKn5K8Q9JBHWJsuMCcFdXHlZVIsF96JQvDdpXIOk0uAInIAcMElKIE7UAYVgMEjeAZv4N16sl6sD+szaU1Z85lD8Kes6Q+gRaFf</latexit>

∂tρ + (v · r)ρ = ρ(r · v)

<latexit sha1_base64="2/YaNgbeAKbqPkOxZcXziK1Gp3E=">ACKXicbVDLSgMxFM34rPVdamLYBEqYpmpC10oFNy4rGC10BnKnTS1oZlkSDKFUvwG/8KNf+HajYKibv0G96YzXaj1QODknHu5954w5kwb1313pqZnZufmcwv5xaXldXC2vqlokitE4kl6oRgqacCVo3zHDaiBWFKOT0KuydjvyrPlWaSXFhBjENIrgWrMIGCu1ClU/BmUY8JbBvupKvIdLfeyTtrR/ASGH3VQ+wfuZX8rUcUl/t1UoumU3BZ4k3pgUq1u3funr4bWKjz7bUmSiApDOGjd9NzYBMPRFoTm7yfaBoD6cE1bVoqIKI6GKaX3uAdq7RxRyr7hMGp+rNjCJHWgyi0lRGYrv7rjcT/vGZiOkfBkIk4MVSQbFAn4dhIPIoNt5mixPCBJUAUs7ti0gUFxNhw8zYE7+/Jk+SyUvYOypVzm8YxypBDm2gblZCHDlEVnaEaqiOC7tAjekGvzr3z5Lw5H1nplDPu2UC/4Hx+A8YyqCg=</latexit>

Dρ Dt = ρ(r · v)

<latexit sha1_base64="N/UYpNiO3RO4TezZtKAqfek5wM=">ACEnicbVDLSgMxFM3UV62vUZdugkXRhWmLnShULALlxWsFjqlZNKMDc0kQ3JHKEO/wY2/4kZQEbeu3PkP/oJg2rQ6oELJ+fcS+49YSK4Ac97d3JT0zOzc/n5wsLi0vKu7p2YVSqKatTJZRuhMQwSWrAwfBGolmJA4Fuwx7J0P/8pw5U8h37CWjG5kjzilICV2u5uEGlCsyoOdFcNsioM8DHeGz52AklCQXBAOwrw9W7bLXolbwT8l/jfpFjZbzx8bnsftb7FnQUTWMmgQpiTNP3EmhlRAOng0KQWpYQmiPXLGmpZLEzLSy0UkDvGWVDo6UtiUBj9SfExmJjenHoe2MCXTNpDcU/OaKUSHrYzLJAUm6fijKBUYFB7mgztcMwqibwmhmtdMe0SmxHYFAs2BH/y5L/kolzy90vlM5vGERojzbQJtpBPjpAFXSKaqiOKLpBd+gRPTm3zr3z7LyMW3PO98w6+gXn9Qt3l6Bl</latexit>

Continuity equation

SLIDE 24

Momentum equation

24

NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

24

What accelerations act on a fluid element?

a = rp ρ

<latexit sha1_base64="5BusnYQRkLHy9+23o/CJtiaqPC8=">ACEXicbVA9SwNBEN2L3/Hr1NJmUYQ0hrtYmEJBsLFUMImQC2Fus5cs2ds9dveEcFwpdjY2/hAbC0UstbOz8L+4Sw0+mDZx3szMwLE8608bwPpzA1PTM7N79QXFxaXl19brWqaK0BqRXKqLEDTlTNCaYbTi0RiENOG2H/eOg3LqnSTIpzM0hoK4auYBEjYKzUdktBKHlHD2L7ZDjQ7yLg0gByQIBIQec5FmgejJvu9te2RsB/yX+N9k+qr5eXd9+3p23fegI0kaU2EIB62bvpeYVgbKMJpXgxSTRMgfejSpqUCYqpb2eiHO9YpYMjqewTBo/Unx0ZxHq4tK2MwfT0pDcU/OaqYmqrYyJDVUkPGgKOXYSDyMB3eYosTwgSVAFLO7YtIDm4exIRZtCP7kyX9JvVL298qVM5vGARpjHm2iLVRCPtpHR+gEnaIaIugG3aNH9OTcOg/Os/MyLi043z0b6Becty9IF6G4</latexit>

This gives us the Euler equations

Dv Dt = rp ρ

<latexit sha1_base64="7/CMVkFBe0iK5ol/wVa2ZSvlZqk=">ACHXicbVC7SgNBFJ2NrxhfUubwSDYGHajYAqFgCksI5gHZEOYncwmQ2ZnlpnZQFi2FD/ARmz8FRsLg1jYiIX/4iSx0MQDwxzOuZd7/FCRpW27U8rtbC4tLySXs2srW9sbmW3d2pKRBKTKhZMyIaHFGUk6qmpFGKAkKPEbqXv9i7NcHRCoq+LUehqQVoC6nPsVIG6mdPXF9iXBchq4nWEcNA/PFgySJyzo5h0dwarsceQzBMIld2RNJO5uz8/YEcJ4PyRXKo5ubu+7ivt7LvbETgKCNeYIaWajh3qVoykpiRJONGioQI91GXNA3lKCqFU+uS+CBUTrQF9I8ruFE/d0Ro0CNFzeVAdI9NeuNxf+8ZqT9YiumPIw04Xg6yI8Y1AKOo4IdKgnWbGgIwpKaXSHuIZOHNoFmTAjO7MnzpFbIO8f5wpVJ4wxMkQZ7YB8cAgecghK4BVQBRg8gCfwAkbWo/VsvVpv09KU9dOzC/7A+vgGbD+nKQ=</latexit>

∂ρv ∂t + r · (ρvv + P · I) = 0

<latexit sha1_base64="+QtjLXmZJmLNpsk0LaWvmZ0v/jA=">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</latexit>

r

SLIDE 25

Momentum equation

25

NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

25

What accelerations act on a fluid element?

a = rp ρ

<latexit sha1_base64="5BusnYQRkLHy9+23o/CJtiaqPC8=">ACEXicbVA9SwNBEN2L3/Hr1NJmUYQ0hrtYmEJBsLFUMImQC2Fus5cs2ds9dveEcFwpdjY2/hAbC0UstbOz8L+4Sw0+mDZx3szMwLE8608bwPpzA1PTM7N79QXFxaXl19brWqaK0BqRXKqLEDTlTNCaYbTi0RiENOG2H/eOg3LqnSTIpzM0hoK4auYBEjYKzUdktBKHlHD2L7ZDjQ7yLg0gByQIBIQec5FmgejJvu9te2RsB/yX+N9k+qr5eXd9+3p23fegI0kaU2EIB62bvpeYVgbKMJpXgxSTRMgfejSpqUCYqpb2eiHO9YpYMjqewTBo/Unx0ZxHq4tK2MwfT0pDcU/OaqYmqrYyJDVUkPGgKOXYSDyMB3eYosTwgSVAFLO7YtIDm4exIRZtCP7kyX9JvVL298qVM5vGARpjHm2iLVRCPtpHR+gEnaIaIugG3aNH9OTcOg/Os/MyLi043z0b6Becty9IF6G4</latexit>

This gives us the Euler equations

Dv Dt = rp ρ

<latexit sha1_base64="7/CMVkFBe0iK5ol/wVa2ZSvlZqk=">ACHXicbVC7SgNBFJ2NrxhfUubwSDYGHajYAqFgCksI5gHZEOYncwmQ2ZnlpnZQFi2FD/ARmz8FRsLg1jYiIX/4iSx0MQDwxzOuZd7/FCRpW27U8rtbC4tLySXs2srW9sbmW3d2pKRBKTKhZMyIaHFGUk6qmpFGKAkKPEbqXv9i7NcHRCoq+LUehqQVoC6nPsVIG6mdPXF9iXBchq4nWEcNA/PFgySJyzo5h0dwarsceQzBMIld2RNJO5uz8/YEcJ4PyRXKo5ubu+7ivt7LvbETgKCNeYIaWajh3qVoykpiRJONGioQI91GXNA3lKCqFU+uS+CBUTrQF9I8ruFE/d0Ro0CNFzeVAdI9NeuNxf+8ZqT9YiumPIw04Xg6yI8Y1AKOo4IdKgnWbGgIwpKaXSHuIZOHNoFmTAjO7MnzpFbIO8f5wpVJ4wxMkQZ7YB8cAgecghK4BVQBRg8gCfwAkbWo/VsvVpv09KU9dOzC/7A+vgGbD+nKQ=</latexit>

∂ρv ∂t + r · (ρvv + P · I) = 0

<latexit sha1_base64="+QtjLXmZJmLNpsk0LaWvmZ0v/jA=">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</latexit>

r

Your exercise!

SLIDE 26

Internal Energy equation

26

NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

26

From first law of thermodynamics,

d✏ = Tds − pdV,

<latexit sha1_base64="MwAK7Jm+9A4dMYP9dH8ACe04aFk=">ACA3icbVC7SgNBFJ2Nrxhfq4KFNoNBsNCwGwstFAI2lhHygmQJs7OzyeDs7DAzqyxLwMbvsLOxUMTWn7DzA/wPJ49CEw9cOJxzL/fe4wtGlXacLys3N7+wuJRfLqysrq1v2JtbDRUnEpM6jlksWz5ShFO6pqRlpCEhT5jDT9m8uh37wlUtGY13QqiBehHqchxUgbqWvBrBDhKIs5vAC1mCg4DEUMGgcwa5dErOCHCWuBNSrOw07kT6/Vjt2p+dIMZJRLjGDCnVdh2hvQxJTEjg0InUQgfIN6pG0oRxFRXjb6YQAPjBLAMJamuIYj9fdEhiKl0sg3nRHSfTXtDcX/vHaiwzMvo1wkmnA8XhQmDOoYDgOBAZUEa5YagrCk5laI+0girE1sBROCO/3yLGmUS+5JqXxt0jgHY+TBHtgHh8AFp6ACrkAV1AEG9+AJvIBX68F6t6s93FrzprMbIM/sD5+AyfmN4=</latexit>

dV = d ✓1 ρ ◆ = −dρ ρ2

<latexit sha1_base64="aSisVqrOfkBmaJbk7BiA9l6N+wI=">ACJnicbVDLSsNAFJ34rPVdelmsAh1YUnqQlELBV24rGBboalMpm0QycPZm6ErL3P7rxJ/wANy4qIu78FKdpF74ODBzOZc79ziR4ApM8OYm19YXFrOreRX19Y3Ngtb20VxpKyBg1FKG8dopjgAWsAB8FuI8mI7wjWcgYXE791z6TiYXADw4h1fNILuMcpAS1C1W3iavYxbZgHpSw7UlCEytNbNkPU2xL3uvDgU5MjUPsZkZm31XSbqFols0M+C+xZqRYO316OBtdVuvdwth2Qxr7LAqiFJty4ygkxAJnAqW5u1YsYjQAemxtqYB8ZnqJNmZKd7Xiou9UOoXAM7U7xMJ8ZUa+o5O+gT6rc3Ef/z2jF4J52EB1EMLKDTRV4sMIR40hl2uWQUxFATQiXf8W0T3QhoJvN6xKs3yf/Jc1K2ToqV651G+doihzaRXuohCx0jGroCtVRA1E0Qs9ojF6NR+PFeDPep9E5Yzazg37A+PwCjOynoA=</latexit>

d✏ ≡ d ✓u ⇢ ◆ = du ⇢ − ud⇢ ⇢2 = −pdV = pd⇢ ⇢2

<latexit sha1_base64="qoTU6WX9oXMiZ3iIPIvLI9T+yAc=">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</latexit> <latexit sha1_base64="wEWkBcRpBlvUPtU+bpJNGlN5fGs=">AB6HicdZDLSgMxFIYzXmu9V0KEiyCqyEzLW0FwYIbly3YC7RDyaSZNjZzIckIZejSlRsXirj1Kdz6Cu58Bn0IM62Civ4Q+Pn+c8g5x404kwqhV2NufmFxaTmzkl1dW9/YzG1tN2UYC0IbJOShaLtYUs4C2lBMcdqOBMW+y2nLHZ2meuSCsnC4FyNI+r4eBAwjxGsNKqjXi6PzKNKyS6WIDIRKlu2lRq7XCwUoaVJqvzJ89vV3lP9vdbLvXT7IYl9GijCsZQdC0XKSbBQjHA6yXZjSNMRnhAO9oG2KfSaDTuCBJn3ohUK/QMEp/d6RYF/Kse/qSh+rofydpfCvrBMr+IkLIhiRQMy+8iLOVQhTLeGfSYoUXysDSaC6VkhGWKBidK3yeojfG0K/zdN27QKpl1H+eoxmCkDdsE+OAQWKIMqOAM10AEUHANbsGdcWHcGPfGw6x0zvjs2QE/ZDx+AOlGkXM=</latexit>

Du Dt = u + p ρ Dρ Dt = ∂u ∂t + (v · r)u = (u + p)r · v

<latexit sha1_base64="hZjEiDxp9xraCDikmGNj+dzDU/s=">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</latexit>

Combine internal energy equation with the momentum equation

∂ρE ∂t + r · [(ρE + P)v] = 0

<latexit sha1_base64="yKzgJ4V0JWRaeH25bGlk429zCyA=">ACTnicbVFdixMxFM3Urzp+VX305WIRVoQysyu4LwuLIvhYwe4uNEPJZO60YTPJkNxZKMP8Ql/EN3+GLz4opm2oO56IeRwzr3JyUlea+UpSb5Eg2vXb9y8Nbwd37l7/6D0cNHJ942TuJMWm3dWS48amVwRo0ntUORZVrPM3P3/T6QU6r6z5QOsas0osjSqVFBSoxQh56YRseS0cKaGBu5WFt90fgjp4AdyIXAvgsrAEXGNJ8xj2tr1Bnj4Hnltd+HUVtri96MI5armiDOAIkXo3EySTYFV0G6A2O2q+li9JkXVjYVGpJaeD9Pk5qytvckNXYxbzWQp6LJc4DNKJCn7WbODp4FpgCSuvCMgQb9u+JVlS+txo6K0Erf1nryf9p84bKw6xVpm4IjdxeVDYhIwt9tlAoh5L0OgAhnQpeQa5EyJfCD/QhpJefBWc7E/Sg8n+5fj49e7OIbsCXvK9ljKXrFj9o5N2YxJ9pF9Zd/Zj+hT9C36Gf3atg6i3cxj9k8Nhr8B1baxoA=</latexit>

Total Energy equation

E = 1 2v2 + ✏

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Volume per unit mass

SLIDE 27

Summary: Euler equations

∂ρ ∂t + r · (ρv) = 0

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∂ρv ∂t + r · (ρvv + P · I) = 0

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∂ρE ∂t + r · [(ρE + P)v] = 0

<latexit sha1_base64="yKzgJ4V0JWRaeH25bGlk429zCyA=">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</latexit>

ρ

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v

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E = 1 2v2 + ✏

<latexit sha1_base64="jzOFCsiorRPsKtCRoLtEdQaGFuY=">ACB3icbVDLSsNAFJ34rPVdSnIYBEoSR0I1QFMFlBfuAJpbJ9KYdOnkwMymUkJ0bf8WNC0Xc+gvu/BsnbRfaemDgcM653LnHizmTyjS/jYXFpeWV1cJacX1jc2u7tLPbkFEiKNRpxCPR8ogEzkKoK6Y4tGIBJPA4NL3Bde43hyAki8J7NYrBDUgvZD6jRGmpUzq4wZfY8QWhqZWldoaHDzY+wQ7EkvE8UDYr5h4nlhTUkZT1DqlL6cb0SAUFOpGxbZqzclAjFKIes6CQSYkIHpAdtTUMSgHT8R0ZPtJKF/uR0C9UeKz+nkhJIOUo8HQyIKovZ71c/M9rJ8q/cFMWxomCkE4W+QnHKsJ5KbjLBFDFR5oQKpj+K6Z9oktRurqiLsGaPXmeNOyKdVqx787K1atpHQW0jw7RMbLQOaqiW1RDdUTRI3pGr+jNeDJejHfjYxJdMKYze+gPjM8fe36Xzw=</latexit>

p

<latexit sha1_base64="YCRLqY3FYFMPr169IibebwVZHg=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZtIvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYglitbCRtRZmx2ZRsCN7y6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAcIzvMKb8+i8O/Ox6K14OQzx/AHzucP2eM+A=</latexit>

✏

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E

<latexit sha1_base64="oW7OyFl8fZOuY5qYrNXg+guGckE=">AB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BETwmYB6QLGF20puMmZ1dZmaFEPIFXjwo4tVP8ubfOEn2oIkFDUVN91dQSK4Nq7eTW1jc2t/LbhZ3dvf2D4uFRU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWj25nfekKleSwfzDhBP6IDyUPOqLFS/a5XLldw6ySryMlCBDrVf86vZjlkYoDRNU647nJsafUGU4EzgtdFONCWUjOsCOpZJGqP3J/NApObNKn4SxsiUNmau/JyY0nocBbYzomaol72Z+J/XSU147U+4TFKDki0WhakgJiazr0mfK2RGjC2hTHF7K2FDqigzNpuCDcFbfnmVNCtl76JcqV+WqjdZHk4gVM4Bw+uoAr3UIMGMEB4hld4cx6dF+fd+Vi05pxs5hj+wPn8AZo7jM0=</latexit>

u

<latexit sha1_base64="wnFhQNKUlf0HWa+NJZ/ckTw+BdQ=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZtovV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYglitbCRtRZmx2ZRsCN7y6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAcIzvMKb8+i8O/Ox6K14OQzx/AHzucP4vuM/Q=</latexit>

u = ⇢✏

<latexit sha1_base64="HQ2fdqdAXxk7nTiJH7EXYnqzJ/s=">AB+nicbVBNSwMxEM36WetXq0cvwSJ4KrtV0ItQ9OKxgv2A7lKy6Wwbmk2WJKuU2p/ixYMiXv0l3vw3Zts9aOuDgcd7M8zMCxPOtHdb2dldW19Y7OwVdze2d3bL5UPWlqmikKTSi5VJyQaOBPQNMxw6CQKSBxyaIejm8xvP4DSTIp7M04giMlAsIhRYqzUK5VTfIV9NZTYh0QznmkVt+rOgJeJl5MKytHolb78vqRpDMJQTrTuem5iglRhlEO06KfakgIHZEBdC0VJAYdTGanT/GJVfo4ksqWMHim/p6YkFjrcRzazpiYoV70MvE/r5ua6DKYMJGkBgSdL4pSjo3EWQ64zxRQw8eWEKqYvRXTIVGEGptW0YbgLb68TFq1qndWrd2dV+rXeRwFdISO0Sny0AWqo1vUQE1E0SN6Rq/ozXlyXpx352PeuLkM4foD5zPH1Dlk2Y=</latexit>

Continuity equation Momentum equation Energy equation

Gas density Gas velocity Gas pressure Gas specific internal energy Gas specific total energy Gas internal energy

27

NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

27 Stress tensor Unit tensor

SLIDE 28

∂ρv ∂t + r · (ρvv + P · I) = ρrΦ

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Summary: Euler equations + Gravity

∂ρ ∂t + r · (ρv) = 0

<latexit sha1_base64="dUBsrN4bCT5H6ytEHTBdCz2GjaE=">ACNHicbVBNSwMxEM36WetX1aOXYBEUoexWQS9C0YvgpYJVoVvKbDbBrPJkmQLZdkf5cUf4kUED4p49TeYrQU/6kDImzdvmJkXJxp47pPztT0zOzcfGmhvLi0vLJaWVu/0jJVhLaI5FLdBKApZ4K2DOc3iSKQhxweh3cnhb16wFVmklxaYJ7cTQEyxiBIylupVzP1JAMj8BZRhw7Ku+zL9Tk+M97AsIOGCfhNLgnUKB/UDyUA9j+5WzQb6Lj7Fb7laqbs0dBZ4E3hU0Tia3cqDH0qSxlQYwkHrtucmpMVswmnedlPNU2A3EKPti0UEFPdyUZH53jbMiGOpLJPGDxif3ZkEOtiQ6uMwfT131pB/ldrpyY6mRMJKmhgnwNilLrhcSFgzhkihLDhxYAUczuikfrIvG+lyY4P09eRJc1Wvefq1+cVBtnIztKFNtIV2kIcOUQOdoSZqIYLu0CN6Qa/OvfPsvDnvX9IpZ9yzgX6F8/EJBlKqUw=</latexit>

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

v

<latexit sha1_base64="NSH9wM7Ar3PVRHh2MFc4hD/zv8=">AB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsxUQZdFNy4r2Ae0Y8lk0jY0kwxJplKG/ocbF4q49V/c+Tdm2lo64GQwzn3kpMTxJxp47rfTmFtfWNzq7hd2tnd2z8oHx61tEwUoU0iuVSdAGvKmaBNwynVhRHAWctoPxbea3J1RpJsWDmcbUj/BQsAEj2FjpsRdIHupZK90MuXK27VnQOtEi8nFcjR6Je/eqEkSUSFIRxr3fXc2PgpVoYRTmelXqJpjMkYD2nXUoEjqv10nqGzqwSoFU9giD5urvjRHOotmJyNsRnrZy8T/vG5iBtd+ykScGCrI4qFBwpGRKsAhUxRYvjUEkwUs1kRGWGFibFlWwJ3vKXV0mrVvUuqrX7y0r9Jq+jCdwCufgwRXU4Q4a0AQCp7hFd6cJ+fFeXc+FqMFJ985hj9wPn8AUZCTCw=</latexit>

E = 1 2v2 + ✏

<latexit sha1_base64="jzOFCsiorRPsKtCRoLtEdQaGFuY=">ACB3icbVDLSsNAFJ34rPVdSnIYBEoSR0I1QFMFlBfuAJpbJ9KYdOnkwMymUkJ0bf8WNC0Xc+gvu/BsnbRfaemDgcM653LnHizmTyjS/jYXFpeWV1cJacX1jc2u7tLPbkFEiKNRpxCPR8ogEzkKoK6Y4tGIBJPA4NL3Bde43hyAki8J7NYrBDUgvZD6jRGmpUzq4wZfY8QWhqZWldoaHDzY+wQ7EkvE8UDYr5h4nlhTUkZT1DqlL6cb0SAUFOpGxbZqzclAjFKIes6CQSYkIHpAdtTUMSgHT8R0ZPtJKF/uR0C9UeKz+nkhJIOUo8HQyIKovZ71c/M9rJ8q/cFMWxomCkE4W+QnHKsJ5KbjLBFDFR5oQKpj+K6Z9oktRurqiLsGaPXmeNOyKdVqx787K1atpHQW0jw7RMbLQOaqiW1RDdUTRI3pGr+jNeDJejHfjYxJdMKYze+gPjM8fe36Xzw=</latexit>

p

<latexit sha1_base64="YCRLqY3FYFMPr169IibebwVZHg=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZtIvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYglitbCRtRZmx2ZRsCN7y6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAcIzvMKb8+i8O/Ox6K14OQzx/AHzucP2eM+A=</latexit>

✏

<latexit sha1_base64="S940j+PXnpyPoS6EHFdj/RxnHVw=">AB73icbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2k3bpZhN3N0IJ/RNePCji1b/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHsHM0nQj+hQ8pAzaqzU6WGiuYhlv1xq+4cZJV4OalAjka/NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQlYzrErqWSRqj9bH7vlJxZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULFojAVxMRk9jwZcIXMiIklClubyVsRBVlxkZUsiF4y+vklat6l1Ua/eXlfpNHkcRTuAUzsGDK6jDHTSgCQwEPMrvDmPzovz7nwsWgtOPnMf+B8/gBOH5Ak</latexit>

E

<latexit sha1_base64="oW7OyFl8fZOuY5qYrNXg+guGckE=">AB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BETwmYB6QLGF20puMmZ1dZmaFEPIFXjwo4tVP8ubfOEn2oIkFDUVN91dQSK4Nq7eTW1jc2t/LbhZ3dvf2D4uFRU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWj25nfekKleSwfzDhBP6IDyUPOqLFS/a5XLldw6ySryMlCBDrVf86vZjlkYoDRNU647nJsafUGU4EzgtdFONCWUjOsCOpZJGqP3J/NApObNKn4SxsiUNmau/JyY0nocBbYzomaol72Z+J/XSU147U+4TFKDki0WhakgJiazr0mfK2RGjC2hTHF7K2FDqigzNpuCDcFbfnmVNCtl76JcqV+WqjdZHk4gVM4Bw+uoAr3UIMGMEB4hld4cx6dF+fd+Vi05pxs5hj+wPn8AZo7jM0=</latexit>

u

<latexit sha1_base64="wnFhQNKUlf0HWa+NJZ/ckTw+BdQ=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZtovV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYglitbCRtRZmx2ZRsCN7y6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAcIzvMKb8+i8O/Ox6K14OQzx/AHzucP4vuM/Q=</latexit>

u = ⇢✏

<latexit sha1_base64="HQ2fdqdAXxk7nTiJH7EXYnqzJ/s=">AB+nicbVBNSwMxEM36WetXq0cvwSJ4KrtV0ItQ9OKxgv2A7lKy6Wwbmk2WJKuU2p/ixYMiXv0l3vw3Zts9aOuDgcd7M8zMCxPOtHdb2dldW19Y7OwVdze2d3bL5UPWlqmikKTSi5VJyQaOBPQNMxw6CQKSBxyaIejm8xvP4DSTIp7M04giMlAsIhRYqzUK5VTfIV9NZTYh0QznmkVt+rOgJeJl5MKytHolb78vqRpDMJQTrTuem5iglRhlEO06KfakgIHZEBdC0VJAYdTGanT/GJVfo4ksqWMHim/p6YkFjrcRzazpiYoV70MvE/r5ua6DKYMJGkBgSdL4pSjo3EWQ64zxRQw8eWEKqYvRXTIVGEGptW0YbgLb68TFq1qndWrd2dV+rXeRwFdISO0Sny0AWqo1vUQE1E0SN6Rq/ozXlyXpx352PeuLkM4foD5zPH1Dlk2Y=</latexit>

Continuity equation Momentum equation Energy equation

Gas density Gas velocity Gas pressure Gas specific internal energy Gas specific total energy Gas internal energy Gravitational Potential

∂ρE ∂t + r · [(ρE + P)v] = ρv · rΦ

<latexit sha1_base64="SF0qiUpFMskgfx859uLbpK17e0=">ACc3icbVFNa9wEJXdr9T92rbQSw4dsi2klCx2GkgvhdBS6HEL3SwMosy2sRWTLSOLAY/4H+vN76L3rpvfLa0DTpgNDjzZun0UxWK+kwjn8G4a3bd+7e27kfPXj46PGTydNnp840losFN8rY84w5oaQWC5SoxHltBasyJc6yi09/uxSWCeN/oabWqQVW2tZSM7QU6vJd1pYxltaM4uSKaC2NPC5+0tgB2+BapYpBpTnBoEqUeAygv1B69PzN0Azo3K3qfwVtZed95HrElOAD3AwmF5RbAWD12g8L2W0mkzjWbwNuAmSEUzJGPV5AfNDW8qoZEr5twyiWtM275xrkQX0caJmvELthZLDzWrhEvb7cw6eO2ZHApj/dEIW/ZqRcsq13frlRXD0l3P9eT/csGi/dpK3XdoNB8eKho/CAN9AuAXFrBUW08YNxK3yvwkvkloF9TP4Tk+pdvgtPDWfJudvj1aHrycRzHDtkle2SfJOSYnJAvZE4WhJNfwYvgZQDB73A3AtfDdIwGuek38iPgDmge7GA=</latexit>

Φ

<latexit sha1_base64="SIWhw4cDOJI1oTS6WRyjqImg48=">AB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lqQY9FLx4r2A9oQ9lsN83S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5QcKZNq7ZQ2Nre2d8q7lb39g8Oj6vFJV8epIrRDYh6rfoA15UzSjmG036iKBYBp71gepf7vSeqNIvlo5kl1Bd4IlnICDa5NGxHbFStuXV3AbROvILUoEB7VP0ajmOSCioN4Vjrgecmxs+wMoxwOq8MU0TKZ4QgeWSiyo9rPFrXN0YZUxCmNlSxq0UH9PZFhoPROB7RTYRHrVy8X/vEFqwhs/YzJDZVkuShMOTIxyh9HY6YoMXxmCSaK2VsRibDCxNh4KjYEb/XldJt1L2reuOhWvdFnGU4QzO4RI8uIYW3EMbOkAgmd4hTdHOC/Ou/OxbC05xcwp/IHz+QPkB4j</latexit>

28

NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

28

r2Φ = 4πGρ

<latexit sha1_base64="LNOBnYisaDXC0SzMRi+oYWDQOs=">ACBHicbVDLSsNAFJ3UV62vqMtuBovgqiS1oBuh6EKXFewDmlgm0kzdDITZiZCV248VfcuFDErR/hzr9x2mahrQcuHM65l3vCRJGlXacb6uwsrq2vlHcLG1t7+zu2fsHbSVSiUkLCyZkN0CKMpJS1PNSDeRBMUBI51gdDX1Ow9EKir4nR4nxI/RkNOQYqSN1LfLHkcBQ/c16DUjCi9g3UsovIaejETfrjhVZwa4TNycVECOZt/+8gYCpzHhGjOkVM91Eu1nSGqKGZmUvFSRBOERGpKeoRzFRPnZ7IkJPDbKAIZCmuIaztTfExmKlRrHgemMkY7UojcV/N6qQ7P/YzyJNWE4/miMGVQCzhNBA6oJFizsSEIS2puhThCEmFtciuZENzFl5dJu1Z1T6u123qlcZnHUQRlcAROgAvOQAPcgCZoAQwewTN4BW/Wk/VivVsf89aClc8cgj+wPn8A52CWVg=</latexit>

Poisson equation

SLIDE 29

Summary: Lagrangian form

ρ

<latexit sha1_base64="94wfdLRBjCkyTC3Y5p5yTs2egQ=">AB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHoxWME84BkCbOT2eyQeSwzs0I+QUvHhTx6g9582+cTfagiQUNRVU3V1Rypmxv/tra1vbG5tl3bKu3v7B4eVo+O2UZkmtEUV7obYUM5k7RlmeW0m2qKRcRpJxrf5X7niWrDlHy0k5SGAo8kixnBNpf6OlGDStWv+XOgVRIUpAoFmoPKV3+oSCaotIRjY3qBn9pwirVlhNZuZ8ZmIyxiPac1RiQU04nd86Q+dOGaJYaVfSorn6e2KhTETEblOgW1ilr1c/M/rZTa+CadMpmlkiwWxRlHVqH8cTRkmhLJ45gopm7FZEa0ysi6fsQgiWX14l7XotuKzVH6qjdsijhKcwhlcQADX0IB7aEILCTwDK/w5gnvxXv3Phata14xcwJ/4H3+ACD6jks=</latexit>

v

<latexit sha1_base64="NSH9wM7Ar3PVRHh2MFc4hD/zv8=">AB9XicbVDLSgMxFL1TX7W+qi7dBIvgqsxUQZdFNy4r2Ae0Y8lk0jY0kwxJplKG/ocbF4q49V/c+Tdm2lo64GQwzn3kpMTxJxp47rfTmFtfWNzq7hd2tnd2z8oHx61tEwUoU0iuVSdAGvKmaBNwynVhRHAWctoPxbea3J1RpJsWDmcbUj/BQsAEj2FjpsRdIHupZK90MuXK27VnQOtEi8nFcjR6Je/eqEkSUSFIRxr3fXc2PgpVoYRTmelXqJpjMkYD2nXUoEjqv10nqGzqwSoFU9giD5urvjRHOotmJyNsRnrZy8T/vG5iBtd+ykScGCrI4qFBwpGRKsAhUxRYvjUEkwUs1kRGWGFibFlWwJ3vKXV0mrVvUuqrX7y0r9Jq+jCdwCufgwRXU4Q4a0AQCp7hFd6cJ+fFeXc+FqMFJ985hj9wPn8AUZCTCw=</latexit>

E = 1 2v2 + ✏

<latexit sha1_base64="jzOFCsiorRPsKtCRoLtEdQaGFuY=">ACB3icbVDLSsNAFJ34rPVdSnIYBEoSR0I1QFMFlBfuAJpbJ9KYdOnkwMymUkJ0bf8WNC0Xc+gvu/BsnbRfaemDgcM653LnHizmTyjS/jYXFpeWV1cJacX1jc2u7tLPbkFEiKNRpxCPR8ogEzkKoK6Y4tGIBJPA4NL3Bde43hyAki8J7NYrBDUgvZD6jRGmpUzq4wZfY8QWhqZWldoaHDzY+wQ7EkvE8UDYr5h4nlhTUkZT1DqlL6cb0SAUFOpGxbZqzclAjFKIes6CQSYkIHpAdtTUMSgHT8R0ZPtJKF/uR0C9UeKz+nkhJIOUo8HQyIKovZ71c/M9rJ8q/cFMWxomCkE4W+QnHKsJ5KbjLBFDFR5oQKpj+K6Z9oktRurqiLsGaPXmeNOyKdVqx787K1atpHQW0jw7RMbLQOaqiW1RDdUTRI3pGr+jNeDJejHfjYxJdMKYze+gPjM8fe36Xzw=</latexit>

p

<latexit sha1_base64="YCRLqY3FYFMPr169IibebwVZHg=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZtIvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYglitbCRtRZmx2ZRsCN7y6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAcIzvMKb8+i8O/Ox6K14OQzx/AHzucP2eM+A=</latexit>

✏

<latexit sha1_base64="S940j+PXnpyPoS6EHFdj/RxnHVw=">AB73icbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2k3bpZhN3N0IJ/RNePCji1b/jzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHsHM0nQj+hQ8pAzaqzU6WGiuYhlv1xq+4cZJV4OalAjka/NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQlYzrErqWSRqj9bH7vlJxZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULFojAVxMRk9jwZcIXMiIklClubyVsRBVlxkZUsiF4y+vklat6l1Ua/eXlfpNHkcRTuAUzsGDK6jDHTSgCQwEPMrvDmPzovz7nwsWgtOPnMf+B8/gBOH5Ak</latexit>

E

<latexit sha1_base64="oW7OyFl8fZOuY5qYrNXg+guGckE=">AB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BETwmYB6QLGF20puMmZ1dZmaFEPIFXjwo4tVP8ubfOEn2oIkFDUVN91dQSK4Nq7eTW1jc2t/LbhZ3dvf2D4uFRU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWj25nfekKleSwfzDhBP6IDyUPOqLFS/a5XLldw6ySryMlCBDrVf86vZjlkYoDRNU647nJsafUGU4EzgtdFONCWUjOsCOpZJGqP3J/NApObNKn4SxsiUNmau/JyY0nocBbYzomaol72Z+J/XSU147U+4TFKDki0WhakgJiazr0mfK2RGjC2hTHF7K2FDqigzNpuCDcFbfnmVNCtl76JcqV+WqjdZHk4gVM4Bw+uoAr3UIMGMEB4hld4cx6dF+fd+Vi05pxs5hj+wPn8AZo7jM0=</latexit>

u

<latexit sha1_base64="wnFhQNKUlf0HWa+NJZ/ckTw+BdQ=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZtovV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYglitbCRtRZmx2ZRsCN7y6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAcIzvMKb8+i8O/Ox6K14OQzx/AHzucP4vuM/Q=</latexit>

u = ⇢✏

<latexit sha1_base64="HQ2fdqdAXxk7nTiJH7EXYnqzJ/s=">AB+nicbVBNSwMxEM36WetXq0cvwSJ4KrtV0ItQ9OKxgv2A7lKy6Wwbmk2WJKuU2p/ixYMiXv0l3vw3Zts9aOuDgcd7M8zMCxPOtHdb2dldW19Y7OwVdze2d3bL5UPWlqmikKTSi5VJyQaOBPQNMxw6CQKSBxyaIejm8xvP4DSTIp7M04giMlAsIhRYqzUK5VTfIV9NZTYh0QznmkVt+rOgJeJl5MKytHolb78vqRpDMJQTrTuem5iglRhlEO06KfakgIHZEBdC0VJAYdTGanT/GJVfo4ksqWMHim/p6YkFjrcRzazpiYoV70MvE/r5ua6DKYMJGkBgSdL4pSjo3EWQ64zxRQw8eWEKqYvRXTIVGEGptW0YbgLb68TFq1qndWrd2dV+rXeRwFdISO0Sny0AWqo1vUQE1E0SN6Rq/ozXlyXpx352PeuLkM4foD5zPH1Dlk2Y=</latexit>

Continuity equation Momentum equation Energy equation

Gas density Gas velocity Gas pressure Gas specific internal energy Gas specific total energy Gas internal energy Gravitational Potential

Φ

<latexit sha1_base64="SIWhw4cDOJI1oTS6WRyjqImg48=">AB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lqQY9FLx4r2A9oQ9lsN83S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5QcKZNq7ZQ2Nre2d8q7lb39g8Oj6vFJV8epIrRDYh6rfoA15UzSjmG036iKBYBp71gepf7vSeqNIvlo5kl1Bd4IlnICDa5NGxHbFStuXV3AbROvILUoEB7VP0ajmOSCioN4Vjrgecmxs+wMoxwOq8MU0TKZ4QgeWSiyo9rPFrXN0YZUxCmNlSxq0UH9PZFhoPROB7RTYRHrVy8X/vEFqwhs/YzJDZVkuShMOTIxyh9HY6YoMXxmCSaK2VsRibDCxNh4KjYEb/XldJt1L2reuOhWvdFnGU4QzO4RI8uIYW3EMbOkAgmd4hTdHOC/Ou/OxbC05xcwp/IHz+QPkB4j</latexit>

29

NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

29

Dρ Dt = ρr · v

<latexit sha1_base64="XWGg/YIP960Xj+XSUftyMCGQ+zw=">ACEnicbVDLSgMxFM3UV62vqks3QVF0YZmpC10oFHThUsFqoTOUTJpQzPJkNwRyjDf4MZfcSOoiFtX7vwHf0EwbV1o9cCFk3PuJfeMBHcgOu+O4WJyanpmeJsaW5+YXGpvLxyaVSqKatTJZRuhMQwSWrAwfBGolmJA4Fuwp7xwP/6pw5W8gH7Cgph0JI84JWClVnHjzSh2Qn2dVfl2Qnk+AjvDl/YlyQUBPu0rQBft8obsUdAv8l3jfZqO01Hj63I+zVvnNbyuaxkwCFcSYpucmEGREA6eC5SU/NSwhtEc6rGmpJDEzQTY8KcebVmnjSGlbEvBQ/TmRkdiYfhzazphA14x7A/E/r5lCdBkXCYpMElH0WpwKDwIB/c5pREH1LCNXc7opl9iMwKZYsiF4yf/JZfVirdXqZ7bNA7RCEW0htbRNvLQPqhU3SG6oiG3SHtGTc+vcO8/Oy6i14HzPrKJfcF6/AFjgoFQ=</latexit>

Dv Dt = rp ρ rΦ

<latexit sha1_base64="KMaxiNV+HTEsLcbDK8vQA3d107M=">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</latexit>

Du Dt = (u + p)r · v

<latexit sha1_base64="VvmBZsbkZA95ka5N4zIxg/Byu6Y=">ACEHicbVDLSsNAFJ3UV62vqEs3g8UXYknahS4UBLtwWcFqoQlMp3YoZNJmLkplNBPcOvuHGhiFuX7vwHf0Fw2rQ6oELh3Pu5d57gkRwDY7zbuWmpmdm5/LzhYXFpeUVe3XtSsepoqxOYxGrRkA0E1yOnAQrJEoRqJAsOugezb0r3tMaR7LS+gnzI/IjeQhpwSM1LJ3vFARmlVxOsiqMAn+ADvpvJHvYkCQTBHm3HgHstu+iUnBHwX+J+k+JpfH0ue181Fr2m9eOaRoxCVQrZuk4CfEQWcCjYoeKlmCaFdcsOahkoSMe1no4cGeMsobRzGypQEPFJ/TmQk0rofBaYzItDRk95Q/M9rphAe+RmXSQpM0vGiMBUYjxMB7e5YhRE3xBCFTe3YtohJiEwGRZMCO7ky3/JVbnkVkrlC5PGMRojzbQJtpFLjpEp+gc1VAdUXSL7tEjerLurAfr2XoZt+as75l19AvW6xevmp7M</latexit>

SLIDE 30

Equation of State (EoS)

Need an Equation of State to close system

f equations.

P = (γ − 1)u = nkBT

<latexit sha1_base64="eOZN7/QnC+5wlAF9e3aRLmbmYIw=">AC3icbVC7SgNBFJ2Nrxhfq5Y2Q4IQC8NuFLQRQmwsI+QF2WZnUySITOzy8ysEJb0Nv6KjYUitv6AnX/jJNlCEw9cOJxzL/feE8aMKu0431ZubX1jcyu/XdjZ3ds/sA+P2ipKJCYtHLFIdkOkCKOCtDTVjHRjSRAPGemE49uZ3kgUtFINPUkJj5HQ0EHFCNtpMAuNuANLHtDxDmC5+4ZTKARBwHqSc5rE9hM7BLTsWZA64SNyMlkKER2F9eP8IJ0JjhpTquU6s/RJTEj04KXKBIjPEZD0jNUIE6Un85/mcJTo/ThIJKmhIZz9fdEirhSEx6aTo70SC17M/E/r5fowbWfUhEnmgi8WDRIGNQRnAUD+1QSrNnEIQlNbdCPEISYW3iK5gQ3OWXV0m7WnEvKtX7y1KtnsWRByegCMrABVegBu5A7QABo/gGbyCN+vJerHerY9Fa87KZo7BH1ifP4Ool5E=</latexit>

For ideal gas,

n = ρ/m

<latexit sha1_base64="qtGP7GEczl1KYbaQhXwc1aESDW8=">AB83icbVDLSgNBEOz1GeMr6tHLYBA8xd0o6EUIevEYwTwgu4TZyWwyZB7LzKwQn7DiwdFvPoz3vwbJ8keNLGgoajqprsrTjkz1ve/vZXVtfWNzcJWcXtnd2+/dHDYNCrThDaI4kq3Y2woZ5I2LOctlNsYg5bcXDu6nfeqLaMCUf7SilkcB9yRJGsHVSKNENCvVAoXMkuqWyX/FnQMskyEkZctS7pa+wp0gmqLSEY2M6gZ/aIy1ZYTSTHMDE0xGeI+7TgqsaAmGs9unqBTp/RQorQradFM/T0xsKYkYhdp8B2YBa9qfif18lsch2NmUwzSyWZL0oyjqxC0wBQj2lKLB85golm7lZEBlhjYl1MRdCsPjyMmlWK8FpfpwWa7d5nEU4BhO4AwCuIa3EMdGkAghWd4hTcv8168d+9j3ri5TNH8Afe5w8K65Bi</latexit>

T

<latexit sha1_base64="wuUeoCr40AnIpr753sXM0828Vs=">AB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BLx4TyAuSJcxOepMxs7PLzKwQr7AiwdFvPpJ3vwbJ8keNLGgoajqprsrSATXxnW/ndzG5tb2Tn63sLd/cHhUPD5p6ThVDJsFrHqBFSj4BKbhuBnUQhjQKB7WB8P/fbT6g0j2XDTBL0IzqUPOSMGivVG/1iyS27C5B14mWkBlq/eJXbxCzNEJpmKBadz03Mf6UKsOZwFmhl2pMKBvTIXYtlTRC7U8Xh87IhVUGJIyVLWnIQv09MaWR1pMosJ0RNSO96s3F/7xuasJbf8plkhqUbLkoTAUxMZl/TQZcITNiYglitbCRtRZmx2RsCN7qy+ukVSl7V+VK/bpUvcviyMZnMleHADVXiAGjSBAcIzvMKb8+i8O/Ox7I152Qzp/AHzucPsPeM3A=</latexit>

kB

<latexit sha1_base64="ul8eZvR3x12+KNVYgJMyL7K3anM=">AB8HicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6DHEi8cI5iHJEmYns8mQeSwzs0JY8hVePCji1c/x5t84SfagiQUNRVU3V1Rwpmxv/tra1vbG5tF3aKu3v7B4elo+OWUakmtEkUV7oTYUM5k7RpmeW0k2iKRcRpOxrfzvz2E9WGKflgJwkNBR5KFjOCrZMex/2spwWqT/ulsl/x50CrJMhJGXI0+qWv3kCRVFBpCcfGdAM/sWGtWE02mxlxqaYDLGQ9p1VGJBTZjND56ic6cMUKy0K2nRXP09kWFhzERErlNgOzL3kz8z+umNr4JMyaT1FJFovilCOr0Ox7NGCaEsnjmCimbsVkRHWmFiXUdGFECy/vEpa1UpwWaneX5Vr9TyOApzCGVxANdQgztoQBMICHiGV3jztPfivXsfi9Y1L585gT/wPn8Ag4WQNw=</latexit>

γ = Cv/Cp

<latexit sha1_base64="wAL2V/u2MDADs8pDmW4L0nr6jxA=">AB+3icbVDLSsNAFL3xWesr1qWbwSK4qkVdCMUu3FZwT6gDWEynbRDZ5IwMymW0F9x40IRt/6IO/GaZuFth64l8M59zJ3TpBwprTjfFtr6xubW9uFneLu3v7BoX1Uaqk4lYQ2Scxj2QmwopxFtKmZ5rSTSIpFwGk7GNVnfntMpWJx9KgnCfUEHkQsZARrI/l2qTfAQmB0i+r+GF2Ynvh2ak4c6BV4uakDkav3V68ckFTShGOluq6TaC/DUjPC6bTYSxVNMBnhAe0aGmFBlZfNb5+iM6P0URhLU5FGc/X3RoaFUhMRmEmB9VAtezPxP6+b6vDGy1iUpJpGZPFQmHKkYzQLAvWZpETziSGYSGZuRWSIJSbaxFU0IbjLX14lrWrFvaxUH67Ktbs8jgKcwCmcgwvXUIN7aEATCDzBM7zCmzW1Xqx362MxumblO8fwB9bnD5Uktg=</latexit>

Number density Temperature Boltzmann constant Adiabatic index

Isothermal EoS:
Polytropic EoS:

γ = 1

<latexit sha1_base64="EXuTpgLwT1Fyd1lq0N1a4yzTmlE=">AB8XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6EUIevEYwTwWULvZJIMmZldZmaFEPIXjwo4tW/8ebfOEn2oIkFDUVN91dUSK4sb7/7a2srq1vbOa28ts7u3v7hYPDuolTVmNxiLWzQgNE1yxmuVWsGaiGcpIsEY0vJ36jSemDY/Vgx0lLJTYV7zHKVonPb7KCWSaxJ0CkW/5M9AlkmQkSJkqHYKX+1uTFPJlKUCjWkFfmLDMWrLqWCTfDs1LE6xD5rOapQMhOZxdPyKlTuqQXa1fKkpn6e2KM0piRjFynRDswi95U/M9rpbZ3FY65SlLFJ0v6qWC2JhM3ydrhm1YuQIUs3drYQOUCO1LqS8CyFYfHmZ1Mul4LxUvr8oVm6yOHJwDCdwBgFcQgXuoAo1oKDgGV7hzTPei/fufcxbV7xs5gj+wPv8AS5Jj+8=</latexit>

P = Kργ

<latexit sha1_base64="eQ6/ufHGfmgA/TtXMTMLizFAYrE=">AB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0I1QdCO4qWAf0MQymU7bofMIMxMhPorblwo4tYPcefOG2z0NYDFw7n3Mu90Qxo9p43rdTWFldW98obpa2tnd29z9g5aWicKkiSWTqhMhTRgVpGmoYaQTK4J4xEg7Gl9P/fYjUZpKcW/SmIQcDQUdUIyMlXpuQEv4S0M1Eg+ZMEQcY4mPbfiVb0Z4DLxc1IBORo9yvoS5xwIgxmSOu78UmzJAyFDMyKQWJjHCYzQkXUsF4kSH2ez4CTy2Sh8OpLIlDJypvycyxLVOeWQ7OTIjvehNxf+8bmIGF2FGRZwYIvB80SBh0Eg4TQL2qSLYsNQShBW1t0I8QgphY/Mq2RD8xZeXSatW9U+rtbuzSv0qj6MIDsEROAE+OAd1cAMaoAkwSMEzeAVvzpPz4rw7H/PWgpPlMEfOJ8/a/KT/w=</latexit>

P = ρc2

s

<latexit sha1_base64="u+kl/NYXWu6FvUj0cf7aw1uDF8=">AB9XicbVBNSwMxEJ2tX7V+VT16CRbBU9mtBb0IRS8eK9gPaLclm2b0GyJFmlLP0fXjwo4tX/4s1/Y9ruQVsfDzem2FmXhBzpo3rfju5tfWNza38dmFnd2/oHh41NQyUYQ2iORStQOsKWeCNgwznLZjRXEUcNoKxrczv/VIlWZSPJhJTP0IDwULGcHGSr06ukZdNZKI9HWv0i+W3LI7B1olXkZKkKHeL351B5IkERWGcKx1x3Nj46dYGUY4nRa6iaYxJmM8pB1LBY6o9tP51VN0ZpUBCqWyJQyaq78nUhxpPYkC2xlhM9L3kz8z+skJrzyUybixFBFovChCMj0SwCNGCKEsMnlmCimL0VkRFWmBgbVMG4C2/vEqalbJ3Ua7cV0u1myOPJzAKZyDB5dQgzuoQwMIKHiGV3hznpwX5935WLTmnGzmGP7A+fwBwGWRYQ=</latexit>

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SLIDE 31

Finite Volume Method

See Hsi-Yu Schive’s lecture tomorrow

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31

U n

ij

<latexit sha1_base64="gK7IVhyN3zF/iHcSzEsND8KB1qk=">AB73icbVBNTwIxEJ3FL8Qv1KOXRmLieyiR6JXjxi4gIJrKRbulBpu2vbNSEb/oQXDxrj1b/jzX9jgT0o+JXt6bycy8MOFMG9f9dgorq2vrG8XN0tb2zu5ef+gqeNUEeqTmMeqHWJNOZPUN8xw2k4UxSLktBWOrqd+64kqzWJ5Z8YJDQeSBYxgo2V2n4vYw+Te9krV9yqOwNaJl5OKpCj0St/dfsxSQWVhnCsdcdzExNkWBlGOJ2UuqmCSYjPKAdSyUWVAfZ7N4JOrFKH0WxsiUNmqm/JzIstB6L0HYKbIZ60ZuK/3md1ESXQcZkhoqyXxRlHJkYjR9HvWZosTwsSWYKGZvRWSIFSbGRlSyIXiLy+TZq3qnVrt+eV+lUeRxGO4BhOwYMLqMNMAHAhye4RXenEfnxXl3PuatBSefOYQ/cD5/AD1BkBk=</latexit>

U n

i+1,j

<latexit sha1_base64="14fQIyOPEh8OtAnLv1WvEHG9tpQ=">AB8nicbVBNSwMxEM3Wr1q/qh69BIsgKGW3CnosevFYwW0L27Vk07SNzSZLMiuUpT/DiwdFvPprvPlvTNs9aPXBwO9GWbmRYngBlz3yksLa+srhXSxubW9s75d29plGpsynSijdjohgkvmAwfB2olmJI4Ea0Wj6nfemTacCXvYJywMCYDyfucErBS4HczfuKdPkzuZbdcavuDPgv8XJSQTka3fJnp6doGjMJVBjAs9NIMyIBk4Fm5Q6qWEJoSMyYIGlksTMhNns5Ak+skoP95W2JQHP1J8TGYmNGceR7YwJDM2iNxX/84IU+pdhxmWSApN0vqifCgwKT/HPa4ZBTG2hFDN7a2YDokmFGxKJRuCt/jyX9KsVb2zau32vFK/yuMogN0iI6Rhy5QHd2gBvIRQo9oRf06oDz7Lw57/PWgpP7KNfcD6+AYG7kL8=</latexit>

U n

i−1,j

<latexit sha1_base64="Ik9ksjKjDudGkwQVWgjoWaSGx4=">AB8nicbVBNSwMxEM3Wr1q/qh69BIvgQctuFfRY9OKxgtsWtmvJpmkbm02WZFYoS3+GFw+KePXePfmLZ70OqDgcd7M8zMixLBDbjul1NYWl5ZXSulzY2t7Z3yrt7TaNSTZlPlVC6HRHDBJfMBw6CtRPNSBwJ1opG1O/9ci04UrewThYUwGkvc5JWClwO9m/NQ7eZjcy2654lbdGfBf4uWkgnI0uXPTk/RNGYSqCDGBJ6bQJgRDZwKNil1UsMSQkdkwAJLJYmZCbPZyRN8ZJUe7itSwKeqT8nMhIbM4j2xkTGJpFbyr+5wUp9C/DjMskBSbpfFE/FRgUnv6Pe1wzCmJsCaGa21sxHRJNKNiUSjYEb/Hlv6RZq3pn1drteaV+lcdRAfoEB0jD12gOrpBDeQjihR6Qi/o1QHn2Xlz3uetBSef2Ue/4Hx8A4TPkME=</latexit>

U n

i,j−1

<latexit sha1_base64="s5aJj7/dOVSbFtHIJ9M0a6eiyWU=">AB8nicbVBNSwMxEM3Wr1q/qh69BIvgQctuFfRY9OKxgtsWtmvJpmkbm02WZFYoS3+GFw+KePXePfmLZ70OqDgcd7M8zMixLBDbjul1NYWl5ZXSulzY2t7Z3yrt7TaNSTZlPlVC6HRHDBJfMBw6CtRPNSBwJ1opG1O/9ci04UrewThYUwGkvc5JWClwO9m/OTh1Jvcy2654lbdGfBf4uWkgnI0uXPTk/RNGYSqCDGBJ6bQJgRDZwKNil1UsMSQkdkwAJLJYmZCbPZyRN8ZJUe7itSwKeqT8nMhIbM4j2xkTGJpFbyr+5wUp9C/DjMskBSbpfFE/FRgUnv6Pe1wzCmJsCaGa21sxHRJNKNiUSjYEb/Hlv6RZq3pn1drteaV+lcdRAfoEB0jD12gOrpBDeQjihR6Qi/o1QHn2Xlz3uetBSef2Ue/4Hx8A4U/kME=</latexit>

U n

i,j+1

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Initial conditions of all

hydrodynamics variables at time step n=0 (a given EoS)

Boundary conditions at

simulation boundaries

Evolve hydrodynamics

variables with a time step dt

SLIDE 32

FLASH Code

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SLIDE 33

FLASH Code: summary

3D AMR HD/MHD multi-physics code
Mainly developed at FLASH center at University of

Chicago (http://flash.uchicago.edu)

Contribution from many individual groups
Written principally in Fortran, with some C and Python (>

1 million lines; 25% are comments)

Current release version 4.6.1
> 700 scientists around the world
> 1000 publications

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SLIDE 34

FLASH Code: Requirements

Fortran compiler
MPI - Message Passing Interface
HDF5 - Hierarchical Data Format

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SLIDE 35

FLASH Code: research applications

Thermonuclear flashes
High energy density physics (HEDP)
Fluid—structure interaction
Star formation
Star-star & Star-plantets interactions
Core-collapse supernova
Galaxy-galaxy cluster
Magnetic field amplification
Turbulence
Cosmology
… (more)

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SLIDE 36

FLASH Code: Physic solvers

Hydrodynamics: unspilt PPM, WENO; split PPM; 2T + Radiation
Magneto-hydrodynamics (MHD): unspilt staggered mesh; split 8-wave
Equation of State (EoS): idea gas; degenerate ionized plasma; nuclear,..
Radiation Transfer: multi-group flux-limited diffusion
Diffusion and Conduction: implicit with AMR
Laser Energy Deposition: geometric optics with inverse Bremsstrahlung
Opacity: constant; multi-material tabular
Particles: tracer; massive; sink; charged
Nuclear Burning
Gravity: constant; point mass; planar; self-gravity
Cosmology
Heating and Cooling as source terms
Magnetic Resistivity; Conductivity
Primordial Chemistry

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SLIDE 37

FLASH Code: Physic solvers

Hydrodynamics: unspilt PPM, WENO; split PPM; 2T + Radiation
Magneto-hydrodynamics (MHD): unspilt staggered mesh; split 8-wave
Equation of State (EoS): idea gas; degenerate ionized plasma; nuclear,..
Radiation Transfer: multi-group flux-limited diffusion
Diffusion and Conduction: implicit with AMR
Laser Energy Deposition: geometric optics with inverse Bremsstrahlung
Opacity: constant; multi-material tabular
Particles: tracer; massive; sink; charged
Nuclear Burning
Gravity: constant; point mass; planar; self-gravity
Cosmology
Heating and Cooling as source terms
Magnetic Resistivity; Conductivity
Primordial Chemistry

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SLIDE 38

FLASH Code: Infrastructure

Driver: split; unsplit
Grid: uniform Grid; Adaptive Mesh Refinement (AMR)
GridParticles: Lagrangian framework
GridSolvers: multigrid; multipole; Barnes-Hut Tree; PFFT; direct solvers
IO: Hierarchical Data Format 5 (HDF5); PnetCDF
Multispecies
Runtime parameters
Monitor: MPI Timers; Hooks for TAU

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38 Block structure PARAMESH AMR

SLIDE 39

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39 See the afternoon lectures for data analysis and visualization

SLIDE 40

FLASH Code: research applications

Highly modular (setup.py)

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SLIDE 41

FLASH Code: Simulation Domain

AMR

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NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

41 Leaf blocks (blockID)

SLIDE 42

FLASH Code: Simulation Domain

Blocks

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NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

42

The grid is composed of

blocks

FLASH4: all blocks are of same

size

May cover different fractions of

the physical domain, depending on a block's resolution

Data storage area for each

block reserves space for some layers of guard cells

Size of guard cells can be

changed by putting a line “GUARDCELLS 6" in the Config file. Default is 4.

SLIDE 43

FLASH Code: Variables

Density: DENS_VAR
Velocities: VELX_VAR, VELY_VAR, VELZ_VAR
Temperature: TEMP_VAR
Pressure: PRES_VAR
Specific internal energy: EINT_VAR

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SLIDE 44

Simulations

All sample simulations can be found in

FLASH/source/Simulation/SimulationMain/

A simulation requires the following 5 files:
Config
Makefile:
Simulation_data.F90
Simulation_init.F90
Simulation_initBlock.F90

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SLIDE 45

Design a FLASH simulation

All sample simulations can be found in

FLASH/source/Simulation/SimulationMain/

A simulation requires the following 5 files:
Config
Makefile:
Simulation_data.F90
Simulation_init.F90
Simulation_initBlock.F90

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SLIDE 46

A simple explosion simulation

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SLIDE 47

The initial condition

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NA School: Comp. Astrophysical Fluid Dynamics, Sep 4-6, 2019, IoA, NTHU

47

“reflect” BC. “reflect” BC. “reflect” BC. “reflect” BC. 1 1

ρ = 1

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v = 0

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✏in = 1

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✏out = 10−5

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SLIDE 48

Config

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SLIDE 49

Using a Flash Unit

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REQUIRES physics/EoS
Default: physics/Eos/EosMain/Gamma

SLIDE 50

Makefile

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SLIDE 51

Simulation_data.F90

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SLIDE 52

Simulation_init.F90

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SLIDE 53

Simulation_init.F90

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SLIDE 54

Simulation_initBlock.F90

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SLIDE 55

Simulation_initBlock.F90

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SLIDE 56

Setup the Simulation

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A machine depend Makefile.h

SLIDE 57

Setup the Simulation

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Run a setup script

A simple example A more complicated example

SLIDE 58

Setup the Simulation

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SLIDE 59

Setup shortcuts

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SLIDE 60

Setup shortcuts (continues)

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SLIDE 61

Compile source codes

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An object/ folder will be generated

SLIDE 62

Run Flash

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Requires

the executable file flash4 and the runtime parameter file flash.par

SLIDE 63

Runtime Parameters (flash.par)

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SLIDE 64

Runtime Parameters

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SLIDE 65

Runtime Parameters

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SLIDE 66

Outputs

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my_sim.log
my_sim.dat
my_sim_hdf5_chk_*
my_sim_hdf5_plt_cnt_*

SLIDE 67

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SLIDE 68

Runtime parameters documentation

http://flash.uchicago.edu/site/flashcode/

user_support/rpDoc_4p61.py

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SLIDE 69

FLASH API

http://flash.uchicago.edu/site/flashcode/

user_support/robodoc-FLASH4_4p61/

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SLIDE 70

Happy coding!

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