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with U,(x,t) e U;(x,t) defined by the lower wall velocity and the cavity geometry using the incompressible flow assumption. iy Ty (Ke hee ee a ee ee i eT Ta Me i; Seen (ETS, ecm Mr pare Mnce Comer Ee RU eects meee —nen 18th International Congress of Mechanical Engineering November 6-11, 2005, Ouro Preto, MG characteristic lower wall temperature 7, . The nondimensional parameters are defined as

Table 1 with U,(x,t) e U;(x,t) defined by the lower wall velocity and the cavity geometry using the incompressible flow assumption. iy Ty (Ke hee ee a ee ee i eT Ta Me i; Seen (ETS, ecm Mr pare Mnce Comer Ee RU eects meee —nen 18th International Congress of Mechanical Engineering November 6-11, 2005, Ouro Preto, MG characteristic lower wall temperature 7, . The nondimensional parameters are defined as

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Figure 1. Physical domain, principle dimensions and coordinate system. A sketch of the solution domain is show in Fig. 1, which also shows the Cartesian system of coordinates used and the cavity principle dimensions. The lower wall movement is described by an arbitrary function A (t) . The upper wall openings allow for mass flow between the cavity and the surroundings. Due to the incompressible flow assumption used in the present work, the fluid velocity within the openings depends on the lower wall velocity - h(t) - and the cavity geometry.
Figure 1. Physical domain, principle dimensions and coordinate system. A sketch of the solution domain is show in Fig. 1, which also shows the Cartesian system of coordinates used and the cavity principle dimensions. The lower wall movement is described by an arbitrary function A (t) . The upper wall openings allow for mass flow between the cavity and the surroundings. Due to the incompressible flow assumption used in the present work, the fluid velocity within the openings depends on the lower wall velocity - h(t) - and the cavity geometry.
Boundary conditions are written, for lower wall movement in upward movement a: Neglecting viscous dissipation effects, considering density variation through the Boussinesq approximation and the remaining thermophysical properties as constants, the mixed convection governing equations are written in nondimensional form as
Boundary conditions are written, for lower wall movement in upward movement a: Neglecting viscous dissipation effects, considering density variation through the Boussinesq approximation and the remaining thermophysical properties as constants, the mixed convection governing equations are written in nondimensional form as
Vorticity (@ ) and stream-function (y ) are defined in terms of the cartesian velocity components, respectively, as In order to decouple the solution for the velocity and pressure fields and reduced to number of equation being simultaneously solved, the flow governing equation are rewritten in Vorticity-Stream Function Formulation as where r,, Re, Pr, Gr and Bi are the cavity aspect ratio, the Reynolds, Prandtl, Grashof and Biot numbers, respectively.
Vorticity (@ ) and stream-function (y ) are defined in terms of the cartesian velocity components, respectively, as In order to decouple the solution for the velocity and pressure fields and reduced to number of equation being simultaneously solved, the flow governing equation are rewritten in Vorticity-Stream Function Formulation as where r,, Re, Pr, Gr and Bi are the cavity aspect ratio, the Reynolds, Prandtl, Grashof and Biot numbers, respectively.
where £ is the clustering parameter (Hoffman, 1992). For the transformation described in Eq. (38), the metrics are also analytically obtained and defined as In order to cluster points near the upper cavity wall, where high solution gradients are expected, anothe ransformation of coordinates is introduced. The grid point clustering near the upper wall is obtained by
where £ is the clustering parameter (Hoffman, 1992). For the transformation described in Eq. (38), the metrics are also analytically obtained and defined as In order to cluster points near the upper cavity wall, where high solution gradients are expected, anothe ransformation of coordinates is introduced. The grid point clustering near the upper wall is obtained by
The governing equations — Eqs. (4,19-20) — are rewritten using the transformed variables 7 and 77 and discretized ising the BTCS Finite Difference scheme. The resulting system of algebraic equations is solved by a Gauss-Seidel subroutine with under-relaxation and local error control.
The governing equations — Eqs. (4,19-20) — are rewritten using the transformed variables 7 and 77 and discretized ising the BTCS Finite Difference scheme. The resulting system of algebraic equations is solved by a Gauss-Seidel subroutine with under-relaxation and local error control.
Figure 2. Flow Field for Stream Function - Cycle 1 Further results on the temporal evolution of the temperature field are shown in Fig. 3. During the initial times, a thin thermal boundary layer develops along the cavity walls and the process is diffusion controlled due to the small velocities (Fig.3a). As the lower wall moves upwards, the discharge opening is opened. Initially, the warmer fluid close to the solid boundary leaves the cavity followed by the cooler fluid originally on the cavity center (Fig.3b). As the lower wall moves towards the maximum height, the displacement rate is reduced and diffusion controlled heating of the cavity center is observed (Fig.3c). SE ee ee Ss, eee Se ee Stream function and temperature results are shown in Figs. 2 and 3, respectively, for different time intervals. Results depicted in Figs. 2a and 3a show flow patterns induced by the upward movement of the lower wall and a uniform temperature distribution within the cavity. As the temperature field develops, natural convection effects become stronger. Results in Fig. 2b show the existence of recirculation zones, represented by dashed lines as the lower wall is in downward motion and the admission valve is opened. The incoming and cooler flow is forced downward between the recirculation zones where warmer fluid is contained (Fig.3b). Results in Fig.2c show that the recirculation zones are weakened as the lower wall moves towards the upper wall. Results also show that the temperature of the fluid leaving the cavity through the discharge valve is at a lower temperature than the temperature near the cavity wall. Figures 2d show the increase of the recirculation zones as the lower wall moves downwards with at higher rate than in Fig 2b. When the lower wall changes direction (h =0), admission and discharge openings are close and the resulting flow pattern is shown in Fig. 2e. Two recirculation zones are present and, due to zero mean velocity, no forced convection is
Figure 2. Flow Field for Stream Function - Cycle 1 Further results on the temporal evolution of the temperature field are shown in Fig. 3. During the initial times, a thin thermal boundary layer develops along the cavity walls and the process is diffusion controlled due to the small velocities (Fig.3a). As the lower wall moves upwards, the discharge opening is opened. Initially, the warmer fluid close to the solid boundary leaves the cavity followed by the cooler fluid originally on the cavity center (Fig.3b). As the lower wall moves towards the maximum height, the displacement rate is reduced and diffusion controlled heating of the cavity center is observed (Fig.3c). SE ee ee Ss, eee Se ee Stream function and temperature results are shown in Figs. 2 and 3, respectively, for different time intervals. Results depicted in Figs. 2a and 3a show flow patterns induced by the upward movement of the lower wall and a uniform temperature distribution within the cavity. As the temperature field develops, natural convection effects become stronger. Results in Fig. 2b show the existence of recirculation zones, represented by dashed lines as the lower wall is in downward motion and the admission valve is opened. The incoming and cooler flow is forced downward between the recirculation zones where warmer fluid is contained (Fig.3b). Results in Fig.2c show that the recirculation zones are weakened as the lower wall moves towards the upper wall. Results also show that the temperature of the fluid leaving the cavity through the discharge valve is at a lower temperature than the temperature near the cavity wall. Figures 2d show the increase of the recirculation zones as the lower wall moves downwards with at higher rate than in Fig 2b. When the lower wall changes direction (h =0), admission and discharge openings are close and the resulting flow pattern is shown in Fig. 2e. Two recirculation zones are present and, due to zero mean velocity, no forced convection is
Figure 3. Temperature Field for Mixed Convection, Cycle | in (a)-(f), Cycle 2 in (g)-(1) e Cycle 3 in (m)-(r) the temperature distribution. The recirculation effect on the temperature distribution is observed from the differences between the results depicted in Figs. 3e and 3f.
Figure 3. Temperature Field for Mixed Convection, Cycle | in (a)-(f), Cycle 2 in (g)-(1) e Cycle 3 in (m)-(r) the temperature distribution. The recirculation effect on the temperature distribution is observed from the differences between the results depicted in Figs. 3e and 3f.
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