Modeling Two-Fluid Interfacial Temperatures
In two-fluid problems, where a sharp interface separates the two fluids, it is possible to have a near discontinuity in tangential velocity at the interface. This situation is particularly true in numerical simulations of gas-liquid flows where the size of the computational grid cells is too large to resolve the interfacial viscous boundary layers. Most numerical approximation methods are unable to accurately deal with a rapid change in tangential velocities at such interfaces and instead produce a smooth transition. This smoothing comes from the use of difference approximations that must reach across one or two grid cells for computing gradients in flow quantities, hence at a sharp interface, velocities on one side are blended with velocities on the other side.
In addition to smoothing, reaching across the interface to compute velocity gradients can also create an unstable solution, when large velocities in one phase feed into the other, slower moving phase. Another problem with velocity jumps across interfaces is that they can lead to unphysical changes in the velocity of a slower moving phase when it enters a computational cell that previously contained only a second phase having a larger velocity.
Interface Slip Approach
The usual computational treatment for this two-fluid problem is to introduce separate velocity fields for each fluid component. This is computationally expensive since it is only at interfaces that two velocity values are required at the same location. That is, within each separate fluid, away from an interface, a single velocity is sufficient. In FLOW-3D a simpler treatment, called the slip model, has been developed using a single velocity field to cover both fluids. A key feature of the model is that the velocity in any grid cell containing both fluids is the velocity of the heavier fluid. A special treatment is used to maintain different tangential velocities across an interface without causing smoothing in the velocity gradients normal to the interface.
If thermal energy transport (i.e., temperature) is computed, then a special treatment is also needed to avoid smoothing temperature differences that may exist across an interface. It is the temperature issue that is discussed in this note in connection with liquid fuel behavior in spacecraft fuel tanks.
If useful simulations of a spacecraft fuel management system are to be performed, it is important to accurately simulate a range of physical processes, including heat transfer, phase change, multi-component gases and time-dependent non-inertial accelerations. Among other things, simulations allow engineers to predict the evolution of pressure in a tank caused by motion-induced mixing of gas and liquid that may have been thermally stratified. The use of a single temperature value in a computational cell, however, introduces artificial numerical diffusion of temperature across the interface when it moves during sloshing, effectively resulting in a non-physical transfer of heat between the liquid and gaseous phases.
A correction for this numerical inaccuracy has been achieved using a variation of the velocity-slip treatment. The energy and mass fluxes from a cell containing both liquid and gas into a cell containing only gas are reset assuming that the gas in the donor cell has the temperature and density of the gas the acceptor cell. The temperature and density in a cell that changes from one containing some liquid to one containing only gas are reset with a gas temperature coming from an average of its neighboring pure gas cells, and a new density is computed assuming that the gas pressure remains unchanged. Finally, pressures, temperatures, densities and energies are reset in cells that changefrom containing pure gas to cells containing some liquid, assuming the new temperature is equal to the average temperature of the neighboring cells containing liquid. It also assumes that the new pressure is the average of all the neighboring cells containing only gas.
|Figure 1a. Temperature distribution at t=0.0.||Figure 1b. Temperature distribution at t=15s without the temperature-slip treatment at liquid-gas interfaces.||Figure 1c. Temperature distribution at t=15s with new temperature-slip treatment at liquid-gas interfaces.|
An illustration of the temperature slip addition is given by a simulation of sloshing of cryogenic liquid in a fuel tank. The cylindrical tank has a radius of 14.5 cm with a spherical bottom end cap. The tank is roughly half filled with liquid at a temperature of 77.0°K with vapor filling the remainder of the tank having a temperature of 87.0°K. Normal gravity at –9.8 m/s2 is applied. A periodic (sinusoidal) shaking is started in the horizontal x direction with a period of 1.4s and amplitude of 0.01m. This initial state is illustrated in Fig.1a, which shows the vertical symmetry plane of the tank aligned with the direction of shaking. No phase change or heat transfer with the walls of the tank is allowed. The computed temperature distribution without the temperature-slip modification after about 10.7 shaking oscillations (15s) is shown in Fig. 1b. There is considerable cooling in the gas above the interface, which has a temperature range of about 9.1°K. A simulation using the new temperature slip treatment, shown in Fig.1c, reveals a much more uniform gas temperature whose range is only 1.29°K.
An interesting point about the new treatment is that the computational time for the more accurate simulation with temperature slip is actually a little less than that for the original, less accurate simulation, 3100s versus 3543s.