Changes between Version 166 and Version 167 of FluxLimitedDiffusion
- Timestamp:
- 04/03/13 12:13:52 (12 years ago)
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FluxLimitedDiffusion
v166 v167 3 3 = Physics of Radiation Transfer = 4 4 5 [[CollapsibleStart(Spectral Intensity)]] 6 == Spectral Intensity == 5 7 Typically when we discuss the radiation field we use the spectral intensity [[latex(I \left ( \nu, \mathbf{x}, \Omega \right ) )]] which is a function of frequency, position, and direction. This is very similar to the phase space density used in deriving the fluid equations [[latex(f \left ( \mathbf{x}, \mathbf{v} \right ) )]] except that 6 8 * light always travels at 'c', so the velocity dependence is just a direction dependence. … … 10 12 Going between photon number and energy just involves a factor of [[latex(h \nu)]] and going from energy density to energy flux density just involves a factor of [[latex(c)]] so we have 11 13 12 [[latex(I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c f \left ( \nu, \mathbf{x}, \Omega, \right ) )]]14 [[latex(I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c f \left ( \nu, \mathbf{x}, \Omega, \right ) )]] 13 15 14 16 This can also be seen by considering the differential energy 15 17 16 [[latex(dE = I \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA dt = h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dV)]]18 [[latex(dE = I \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA dt = h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dV)]] 17 19 18 20 where the number of photons traveling normal to the surface dA that cross the surface dA in time dt is just the number of photons in the volume dV = dA c dt (assuming the photons are headed normal to dA)... 19 21 20 22 so we also have 21 [[latex(dE = h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA c dt)]]23 [[latex(dE = h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA c dt)]] 22 24 23 25 which gives 24 26 25 [[latex(I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c f \left ( \nu, \mathbf{x}, \Omega, \right ) )]] 26 27 28 There are also parallels between the radiation transport equation and the Boltzmann transport equation except that photons don't experience body forces, and photon's don't collide with each other, but scatter off of particles. 29 || [[latex(\frac{\partial}{\partial t} I_\nu + c \mathbf{n} \cdot \nabla I_\nu = c \eta_\nu - c \chi_\nu I_\nu)]] || 30 || [[latex(\frac{\partial}{\partial t} f + \mathbf{v} \cdot \nabla f + \mathbf{a} \cdot \nabla_v f = \left ( \frac{\partial f}{\partial t} \right )_{coll} )]] || 27 [[latex(I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c f \left ( \nu, \mathbf{x}, \Omega, \right ) )]] 28 29 30 [[CollapsibleEnd()]] 31 32 [[CollapsibleStart(Deriving the Transport Equation)]] 33 == Deriving the Transport Equation == 34 35 If we consider the Boltzmann transport equation for photons of a specific frequency [[latex(f_\nu)]] we have 36 37 [[latex(\frac{\partial}{\partial t} f_\nu + \mathbf{v} \cdot \nabla f + \mathbf{F} \cdot \nabla_p f = \left ( \frac{\partial f}{\partial t} \right )_{coll} )]] 38 39 Now photons don't experience body forces, always travel at the speed of light, and in general the "collision term" consists of photon emission and absorptions... so we have 40 41 [[latex(\frac{\partial}{\partial t} f_\nu + c \mathbf{n} \cdot \nabla f_\nu = A_{emiss}_\nu - \chi_\nu f_nu c )]] 42 43 where [[latex(A_{emiss}_\nu)]] is the emission rate of photons of frequency [[latex(\nu)]] and the mean free path length is given by [[latex(\chi_\nu = \sigma_nu n)]] where [[latex(\sigma_nu)]] is the particle scattering cross section and [[latex(n)]] is the number density of particles. 44 45 Now if we multiply through by [[latex( h\nu)]] 46 47 we have 48 49 [[latex(\frac{\partial}{c \partial t} I_\nu + \mathbf{n} \cdot \nabla I_\nu = \eta_\nu - \sigma_\nu I_\nu )]] 50 51 where 52 [[latex(\eta_\nu = h\nu A_{emiss}_\nu)]] is the radiative power 31 53 32 54 If we solve the transport equation along a characteristic 33 55 34 [[latex(\left [ \mathbf{x} \left ( s \right ) , t \left ( s \right ) \right ] = \left [ \mathbf{x0} + \mathbf{n} s, \frac{s}{c} \right ] )]]56 [[latex(\left [ \mathbf{x} \left ( s \right ) , t \left ( s \right ) \right ] = \left [ \mathbf{x0} + \mathbf{n} s, \frac{s}{c} \right ] )]] 35 57 36 58 we have 37 59 38 [[latex(\frac{d I_\nu}{d s} = \frac{\partial I_\nu}{\partial x^i} \frac{\partial{x^i}}{\partial s} + \frac{\partial I_\nu}{\partial t}\frac{\partial t}{\partial s} = \mathbf{n} \cdot \nabla I_\nu + \frac{1}{c}\frac{\partial I_\nu}{\partial t} = \eta_\nu(s) - \chi_\nu(s) I_\nu (s) )]]60 [[latex(\frac{d I_\nu}{d s} = \frac{\partial I_\nu}{\partial x^i} \frac{\partial{x^i}}{\partial s} + \frac{\partial I_\nu}{\partial t}\frac{\partial t}{\partial s} = \mathbf{n} \cdot \nabla I_\nu + \frac{1}{c}\frac{\partial I_\nu}{\partial t} = \eta_\nu(s) - \chi_\nu(s) I_\nu (s) )]] 39 61 40 62 where [[latex(f(s) = f(\mathbf{x}(s), t(s)) = f \left ( \mathbf{x_0}+\mathbf{n} s, \frac{s}{c} \right ) )]] … … 42 64 and then we can divide through by [[latex(\chi_\nu(s))]] we get 43 65 44 [[latex(\frac{dI_\nu}{\chi_\nu(s) ds} = \frac{\eta_\nu(s)}{\chi_\nu(s)} - I_\nu(s) = S_\nu(s) - I_\nu(s))]]66 [[latex(\frac{dI_\nu}{\chi_\nu(s) ds} = \frac{\eta_\nu(s)}{\chi_\nu(s)} - I_\nu(s) = S_\nu(s) - I_\nu(s))]] 45 67 46 68 Now if we define 47 69 48 [[latex(d\tau_\nu = \chi_\nu(s) ds )]]70 [[latex(d\tau_\nu = \chi_\nu(s) ds )]] 49 71 which gives 50 72 51 || [[latex(\tau_\nu(s) = \int\limits_0^s \chi_\nu(s') ds')]] || 52 || [[latex(s(\tau_\nu) = \int\limits_0^\tau_\nu \frac{1}{\chi_\nu} d\tau'_\nu )]] || 73 [[latex(\tau_\nu(s) = \int\limits_0^s \chi_\nu(s') ds')]] 74 and 75 [[latex(s(\tau_\nu) = \int\limits_0^\tau_\nu \frac{1}{\chi_\nu} d\tau'_\nu )]] 53 76 54 77 we can write the transport equation in the simplest form 55 78 56 [[latex(\frac{dI_\nu}{d\tau_\nu} = S_\nu(\tau_\nu) - I_\nu(\tau_\nu))]]79 [[latex(\frac{dI_\nu}{d\tau_\nu} = S_\nu(\tau_\nu) - I_\nu(\tau_\nu))]] 57 80 58 81 although the RHS is now more difficult to evaluate as 59 82 60 [[latex( f \left ( \tau_\nu \right ) = f \left ( s \left ( \tau_\nu \right ) \right ) = f \left ( \mathbf{x} \left (s \left ( \tau_\nu \right ) \right ), t \left ( s \left ( \tau_\nu \right ) \right ) \right ) )]]83 [[latex( f \left ( \tau_\nu \right ) = f \left ( s \left ( \tau_\nu \right ) \right ) = f \left ( \mathbf{x} \left (s \left ( \tau_\nu \right ) \right ), t \left ( s \left ( \tau_\nu \right ) \right ) \right ) )]] 61 84 62 85 Also if we include scattering then the source function can depend on the mean radiative flux [[latex(\frac{cE}{4 \pi})]] and the transport equation becomes an integro-differential equation that must be solved iteratively... … … 69 92 || [[latex(\tau >> 1 \mbox{, } \beta \tau >> 1)]] || dynamic diffusion limit || 70 93 94 [[CollapsibleEnd()]] 71 95 72 96 [[CollapsibleStart(Equations of Radiation Hydrodynamics)]]
