R. Betti Lecture 25 Hydrodynamic ScalingR. Betti . Lecture 25 . Hydrodynamic Scaling . 2...

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INTRODUCTION TO INERTIAL CONFINEMENT FUSION R. Betti Lecture 25 Hydrodynamic Scaling

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Transcript of R. Betti Lecture 25 Hydrodynamic ScalingR. Betti . Lecture 25 . Hydrodynamic Scaling . 2...

  • INTRODUCTION TO INERTIAL CONFINEMENT FUSION

    R. Betti

    Lecture 25

    Hydrodynamic Scaling

  • 2

    Hydrodynamic scaling is used to extrapolate OMEGA results to NIF laser energies

  • The Euler equations are scale invariant and depend on a single dimensionless parameter

    0vtρ ρ∂ +∇ =∂

    0/itV Rτ = ˆ / iv v V= 0ˆ /r r R= 0ˆ R∇ = ∇

    0v v v pt

    ρ ∂ + ∇ +∇ = ∂

    / Ap p P=0/ρ ρ ρ=

    2 23 1 5 1 02 2 2 2

    p v v p vt

    ρ ρ∂ + +∇ + = ∂

  • 2 2 2 23 1 5 1ˆˆ ˆˆ ˆ ˆ ˆ ˆ 02 2 2 2

    p Mach v v p Mach vρ ρτ∂ + +∇ + = ∂

    ˆ ˆ ˆ ˆ 0vρ ρτ∂

    +∇ =∂

    2 ˆ ˆ ˆˆ ˆ ˆ ˆ 0vMach v v pρτ∂ + ∇ +∇ = ∂

    22 0 i

    A

    VMachPρ

    =

    • The single dimensionless parameter is the Mach number

    • Dimensionless Euler equations:

  • • Keeping the Mach number fixed results in similar implosions

    3/52 2 22 0

    3/5 2/5~i i iA

    A A A

    V V VPMachP P Pρ

    α α = =

    Hydroequivalent implosions have: • Same implosion velocity • Same ablation pressure • Same adiabat

  • • The stagnation pressure and density is the same for hydroequivalent implosions

    3 4stag AP P Mach

    −=

    1 20stag Machρ ρ

    −=

    3/5

    0 ~ APρα

  • • Since the hydrodynamics is the same, the classical growth of the instabilities is the same for hydroequivalent implosions

    • The classical growth factor is the same

    0

    stag te Growth Factorγηη

    = = −

  • • To achieve the same ablation pressure, hydroequivalent implosions require the same laser intensity or radiation pressure

    2/3~A LP I

    3.5~A radP T

    • To achieve the same implosion velocity, hydroequivalent implosions require absorbed laser energy proportional to mass

    21 ~2 shell i Hydro abs Hydro abs L

    M V E Eη η η=

    • If the absorption fraction is scale invariant then

    ~L shellE M

    Same for hydroequivlent implosions

  • • The ablative stabilization of the RT instability is the same for hydroequivalent implosions

    • The specific ablation rate is the same for hydroequivalent implosions

    1/3~A Lm I

    3~A radm T

    1/31/15 3/5

    3/50

    ~ ~ ~( / )

    A LA L

    A

    m IV IP

    αρ α

  • • Time scale is proportional to radius

    0/itV Rτ =0

    i

    R tV

    τ = 0~t R

    2 30 0~ ~ ~L L LE Power time I R t I R×

    • The laser energy is proportional to volume

    • The laser power is proportional to surface

    2 20 0~ ~LPower I R R

    • The adiabat depends on initial shock strength that depends on initial ablation pressure (i.e. intensity). Keep IL(τ) fixed

  • • Summary of hydroscaling

    1/3L LMass ~ Volume ~ E R~E

    1/3Ltime ~ R ~ E

    20

    ( )I ( ) ~LLaser Power fixed

    Rττ − =

    2 2/3LPower ~ R ~ E

    0/itV Rτ =

    η(0)= Initial Nonuniformities ~ R ~ EL1/3

    • To keep the same relative size of the nonuniformities the same:

    Vi and α are the same

  • 12

    Hydrodynamic scaling of the no-alpha fusion yield

  • 13

    Hydro scaling

    For 2x amplification

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