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Dear all,
I would like to perform numerical simulations of the heat transfer/temperature field in a static bath of superfluid helium. The heat conduction in superfluid helium can be modeled in two regimes depending on the heat flux. For low heat fluxes ##\dot{q}##, the temperature gradient depends linearly on the heat flux (Landau regime). With increasing heat flux, a second term becomes more important (Goerter-Mellink regime), in which the temperature gradient depends non-linearly on the transferred heat flux (the exponent ##m## experimentally was determined to be ##3.4##):
$$grad(T) = \underbrace{- f_{L} \dot{q}}_{Landau} + \underbrace{- f_{GM} \dot{q}^m}_{Goerter-Mellink}$$
Assuming the contribution of the Landau-term negligible (high ##\dot{q}##), after re-arranging the correlation can be plugged in into the diffusion equation (##f_{GM}^{-1} = const.##):
$$\rho c \frac{\partial T}{\partial t} = -\sqrt[m]{f_{GM}^{-1}}\; div\;\sqrt[m]{grad(T)}$$
$$\frac{\partial T}{\partial t} = \underbrace{\frac{-\sqrt[m]{f_{GM}^{-1}}}{\rho c}}_{a_{GM}}\; div\;\sqrt[m]{grad(T)}$$
Exemplarily for the x-direction:
$$\frac{\partial T}{\partial t} = a_{GM}\frac{\partial\left(\frac{\partial T}{\partial x}\right)^{\frac{1}{m}}}{\partial x}$$
Now I would like to discretize the above equation and I obtain
$$\frac{T_i^{n+1}-T_i^{n}}{\Delta t} =a_{GM}\frac{\left(\frac{T_{i+1}-T_{i}}{\Delta x}\right)^{\frac{1}{m}} - \left(\frac{T_{i}-T_{i-1}}{\Delta x}\right)^{\frac{1}{m}}}{\Delta x} $$
$$\frac{T_i^{n+1}-T_i^{n}}{\Delta t} =a_{GM}\frac{\sqrt[m]{T_{i+1}-T_{i}} - \sqrt[m]{T_{i}-T_{i-1}}}{\Delta x^{\frac{m+1}{m}}} $$
$$T_i^{n+1}-T_i^{n} =\underbrace{\frac{a_{GM}\Delta t}{\Delta x^{\frac{m+1}{m}}}}_{M}\left[\sqrt[m]{T_{i+1}-T_{i}} - \sqrt[m]{T_{i}-T_{i-1}}\right] $$
Using the Euler Upwind Scheme:
$$T_i^{n+1} -M\;\sqrt[m]{T_{i+1}^{n+1}-T_{i}^{n+1}} +M\; \sqrt[m]{T_{i}^{n+1}-T_{i-1}^{n+1}}=T_i^{n} $$
This last equation cannot be transformed into an LES for the cells of the numerical mesh. I hope that I'm correct by stating that I need to linearize the equation at a certain point and/or maybe solve it iteratively. Unfortunately I'm not sure, if what I try is even possible at all. I'd appreciate, if someone could guide, help or explain to me if and how this nonlinear correlation can be discretized.
Thanks,
stockzahn
I would like to perform numerical simulations of the heat transfer/temperature field in a static bath of superfluid helium. The heat conduction in superfluid helium can be modeled in two regimes depending on the heat flux. For low heat fluxes ##\dot{q}##, the temperature gradient depends linearly on the heat flux (Landau regime). With increasing heat flux, a second term becomes more important (Goerter-Mellink regime), in which the temperature gradient depends non-linearly on the transferred heat flux (the exponent ##m## experimentally was determined to be ##3.4##):
$$grad(T) = \underbrace{- f_{L} \dot{q}}_{Landau} + \underbrace{- f_{GM} \dot{q}^m}_{Goerter-Mellink}$$
Assuming the contribution of the Landau-term negligible (high ##\dot{q}##), after re-arranging the correlation can be plugged in into the diffusion equation (##f_{GM}^{-1} = const.##):
$$\rho c \frac{\partial T}{\partial t} = -\sqrt[m]{f_{GM}^{-1}}\; div\;\sqrt[m]{grad(T)}$$
$$\frac{\partial T}{\partial t} = \underbrace{\frac{-\sqrt[m]{f_{GM}^{-1}}}{\rho c}}_{a_{GM}}\; div\;\sqrt[m]{grad(T)}$$
Exemplarily for the x-direction:
$$\frac{\partial T}{\partial t} = a_{GM}\frac{\partial\left(\frac{\partial T}{\partial x}\right)^{\frac{1}{m}}}{\partial x}$$
Now I would like to discretize the above equation and I obtain
$$\frac{T_i^{n+1}-T_i^{n}}{\Delta t} =a_{GM}\frac{\left(\frac{T_{i+1}-T_{i}}{\Delta x}\right)^{\frac{1}{m}} - \left(\frac{T_{i}-T_{i-1}}{\Delta x}\right)^{\frac{1}{m}}}{\Delta x} $$
$$\frac{T_i^{n+1}-T_i^{n}}{\Delta t} =a_{GM}\frac{\sqrt[m]{T_{i+1}-T_{i}} - \sqrt[m]{T_{i}-T_{i-1}}}{\Delta x^{\frac{m+1}{m}}} $$
$$T_i^{n+1}-T_i^{n} =\underbrace{\frac{a_{GM}\Delta t}{\Delta x^{\frac{m+1}{m}}}}_{M}\left[\sqrt[m]{T_{i+1}-T_{i}} - \sqrt[m]{T_{i}-T_{i-1}}\right] $$
Using the Euler Upwind Scheme:
$$T_i^{n+1} -M\;\sqrt[m]{T_{i+1}^{n+1}-T_{i}^{n+1}} +M\; \sqrt[m]{T_{i}^{n+1}-T_{i-1}^{n+1}}=T_i^{n} $$
This last equation cannot be transformed into an LES for the cells of the numerical mesh. I hope that I'm correct by stating that I need to linearize the equation at a certain point and/or maybe solve it iteratively. Unfortunately I'm not sure, if what I try is even possible at all. I'd appreciate, if someone could guide, help or explain to me if and how this nonlinear correlation can be discretized.
Thanks,
stockzahn
Last edited: