Zhao and Kurzweg

The Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm for heat transfer and fluid flow problems is extended to time-periodic situations. A vectorized line group method for solving the system of associated algebraic equations in a rectangular two-dimensional computational domain is developed to speed up the computations. A multiblock procedure with the line group method is used to solve a piston-driven oscillating heat transfer problem. The numerical results obtained show some interesting new phenomena and agree with analytical results where such comparisons are possible.

We consider the oscillatory flow configuration shown in Fig. I. It is based on the thermal pump configuration [I71 and consists essentially of an open-ended flat plate channel of length L and width 2a connected to two reservoirs inside of which there are sinusoidally oscillating rectangular piston plates. The space within the channel and reservoirs is filled with an incompressible fluid and the conduit walls are insulated. The left reservoir walls including the piston face are maintained at constant temperature Th while the right reservoir walls and piston face are maintained at Tc. The temperature gradient normal to the conduit axis vanishes at both the channel walls and the channel axis. Both the axial (U) and transverse (V) velocity components vanish on all nonmovable wall sections, and the axial component has a velocity equal to the piston velocity at the piston face. The reservoirs have width 2b. The configuration symmetry is such that the actual numerical calculations for the two-dimensional spatial velocity field and temperature field will need to be done only over the upper half (i.e., y > 0) of the cross section shown in Fig. 1. The fact that the pistons perform in-phase sinusoidal oscillations at an angular frequency ω also allows a reduction of the time portion of the calculations to only the first half of each oscillation period.

In the numerical calculations that extended over the region y > 0, we found it convenient to break the computation region into three subregions by using nonuniform grids as indicated in Fig. 2. All three grids (one time-independent one for the connecting channel and two movable grids for the end reservoirs) had nonuniform grid spacing generated by the stretching formulas found in the book by Anderson [I41 such that those regions near the walls and at the edge of the jet region at y = 1 where large shear regions are anticipated have higher grid density. This generally involved stretching parameters in both the axial and transverse direction for each of the three grids. The grid sizes at the edges of the subregions at nondimensional distance x = Ll2a and x = -L/2a were made equal. They components of the grids were generated at the beginning of each run while the x components had to be generated at each time step because of the moving piston boundaries. Note that the tidal displacement enters the problem only through boundary condition (5) and not through Eqs. (1)-(4).

First, the time-periodic flow indicates the existence of two counterrotating vortexes in each reservoir whose rotation direction remains the same throughout the oscillation cycle. The rotation sense is such that the fluid motion along the dividing line (i.e., x axis) is always toward the pistion wall and away from the connecting channel exit. This type of counterrotating behavior is apparently not only present under laminar conditions but also

has been found experimentally under conditions where the fluid within the end reservoirs is turbulent [16]. Note how a narrow fluid jet shoots across the right reservoir starting at about ωt = v/4 and impinging on the movable piston wall at ωt = 2713. This corresponds to a velocity of about 400 cm/s and is consistent with what can be expected by fluid continuity considerations and the much slower piston face velocity. The typical buildup to the final periodic state of the axial velocity component, for the case of a = 3, Pr = I, and Ax = 6 cm, from its initial value of zero is shown in Fig. 6. It indicates the value of U(x, y, r) at the center of the connecting channel and at x = - 175 (length in multiples of the channel half width a) and y = 0 at the center of the left reservoir. The comparison of the analytical value [6] for the axial velocity existing in a channel of infinite length at the channel center is shown as a comparison with the numerical results for x/L= 0.2, It does confirm the accuracy of the present numerical results. In comparing the analytical and numerical values of U at the connecting channel center. we introduced a phase shift of 17º. Note the sharp periodic velocity pulses occurring in the reservoir at considerable distance from the channel exit.

The temperature distribution corresponding to the streamline pattern shown in Fig. 5 is found in Fig. 7 at 15º intervals during the oscillation cycle. Its main feature is that the temperatures in the reservoirs, not directly along the walls or along the x axis extension of the connecting channel, are close to being isothermal. The isotherms shown in the figure are spaced at T = 0.02 intervals, with the temperature values on the left reservoir walls being T = 1 and those on the right reservoir walls being T = - 1. The average temperature within the left reservoir is approximately T = 0.75 and that in the right reservoir T = -0.75. Note that there is a thermal plume entering the right reservoir approximately coincident with the impinging fluid jet. This corresponds to a heat pulse entering the colder right reservoir and has the counterpart of a cold fluid pulse entering the left reservoir during the earlier part of the oscillation cycle. This type of heat exchange represents the essence of the thermal pumping technique [I7] whereby heat is exchanged between two reservoirs at rates orders of magnitude higher than by pure conduction yet there is no convective mass exchange between the fluids in the reservoirs as long as the tidal displacement remains smaller than about half the channel length. Although some diffusive mass transfer is possible, this will be very small when dealing with liquids generally characterized by very high Schmidt numbers. The fact that the temperatures are almost uniform in the reservoirs during the oscillation cycle clearly indicates that the sinusoidal piston motions provide excellent fluid mixing within the chambers.