ion channel Scale similarity criteria the Rayleigh number an

Scale similarity criteria (the Rayleigh number and the rotation parameter, constructed from the scale quantities) specified in the course of the simulations, were determined for the conjugate and the non-conjugate settings as follows: where ΔT0 is the temperature difference (ΔT0=–). In the conjugate setting, the convection in the cavity is actually determined by the values of the effective Rayleigh number Ra[27] and the effective rotation parameter K (in the case of the rotating cavity), which can be constructed from the difference ΔT between the space- and time-averaged temperatures at the fluid-solid interface. Thus,
The effective values of the governing parameters are computed after the end of the computations carried out in the conjugate setting. To extract the actual conjugate heat exchange effects from the obtained simulation data, the scale similarity criteria are selected (iteratively) such that the effective Rayleigh number is almost equal to some reference value chosen when solving the non-conjugate problem. In this ion channel investigation, the reference value of the Rayleigh number was approximately equal to 106.
Numerical method The finite-volume ‘unstructured’ code SINF/Flag-S, developed by the team of the Fluid Dynamics, Combustion and Heat Transfer department of Peter the Great St. Petersburg Polytechnic University, was used for the main series of computations. As one of the options for solving Navier–Stokes equations, this code uses an original version of the implicit fractional-step method (FSM) to advance in physical time. The FSM is widely used (in various variants) to solve numerically unsteady hydrodynamic problems (see, for example, Refs. [29–31]). The principle of the FSM is in separating the spatial operators forming the momentum equation, and in interpreting the role of the pressure gradient as a projection operator that converts an arbitrary velocity field to a solenoidal one. As in Ref. [29], the below-described method is based on the Crank–Nicolson scheme (which has a second-order accuracy in time) for the momentum equation, approximated as follows: where Δt is the time step; n, n+ 1 are the time layers. We should note that the convective term in Eq. (8) can be computed by different methods. The most common one is extrapolation of the convective term as an ensemble from two preceding time levels by the Adams–Bashforth scheme [29]:
Since this scheme is explicit (with respect to the convective terms), the computational algorithm as a whole is stable only for Courant numbers less than unity. As noted in Ref. [31], using Eq. (9) for flows with high Reynolds numbers may require the introduction of a stabilizing term. Additionally, the experience of using this scheme in the SINF/Flag-S code revealed that for highly skewed cells, the Courant number should be significantly reduced for computational stability. In view of this, the implicit setting is preferable for computing the convective term. In particular, Ref. [30] also suggested, with respect to the Crank–Nicolson scheme, to compute this term by the semi-implicit scheme:
According to Ref. [30], scheme (10) provides computational stability at Courant numbers greater than unity while involving only two time levels. It is known, however, that the Crank–Nicolson scheme is neutrally stable and can lead to non-physical oscillations with time in a number of problems with strong nonlinear (convective) effects. A common way of suppressing these oscillations is adopting a scheme weighing the contributions from the first-order implicit Euler scheme and the Crank–Nicolson scheme. This, however, leads to a reduction in the order of time accuracy. To settle these issues, the SINF/Flag-S code uses an original modification for computing the convective terms, which combines extrapolation from two preceding time levels by the Adams–Bashforth scheme with implicitness introduced into the scheme as follows: and the velocity in the intermediate (n+  1/2) level computed as