Development of immersed boundary methods for isothermal and thermal flows 7

  Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows     Chapter Applications of Developed IBM Solvers to Simulate Three Dimensional Incompressible Thermal Flows In this chapter, the IBM solvers proposed in Chapters 2, and are applied to numerically study three-dimensional incompressible thermal flows. Specifically, the forced convective heat and mass transfer from stationary or rotating spheres in uniform cross flow, and natural convective heat transfer inside spherical annulus are studied. The incompressible viscous thermal flow around spheres is a fundamental fluid dynamic and heat transfer problem with widespread scientific and engineering applications. Bioreactors, industrial fluidized beds, combustion systems, and chemical processes, etc. are among the well-known examples. In spite of its simple and axisymmetric geometry, the sphere always induces fully three-dimensional flows which would admit complicated kinematics. The forced convective heat transfer, as pointed out, is a passive scalar transport governed by the flow field such that the thermal field should experience corresponding variations. The natural convective heat transfer, on the other 215      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows hand, is induced by buoyancy due to the temperature gradients so that the fluid and thermal fields interact intensively and are strongly coupled. In the present study, the fluid behaviors as well as the heat transfer performances for both forced and natural convective heat transfer are numerically investigated. Specifically, forced convective heat and mass transfer around a single stationary sphere, two tandem stationary spheres and a single streamwise rotating sphere, as well as natural convective heat transfer inside concentric and vertically eccentric spherical annuluses are simulated. The thermal conditions on the sphere surface, in most cases, are set to be of Dirichlet type, for the convenience of direct comparison and analysis. Heat flux condition is only considered in a few particular cases.   7.1 Forced convective heat and mass transfer around a stationary isolated sphere Knowledge concerning the flow around an isolated stationary sphere has been well accumulated by experiments and numerical simulations during the past decades. Just as its two-dimensional counterpart (flow around a circular cylinder), the problem of flow around a stationary sphere has been considered as an excellent case for validating new numerical methodologies in three dimensions. The flow field around a single sphere is recognized to enjoy rich transition 216      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows modes, depending on the Reynolds number Re , which is frequently defined as Re = ρU ∞ D based on the sphere diameter D and the uniform incoming μ velocity U ∞ . At low Reynolds numbers, an axisymmetric vortex ring is formed behind the sphere and attached to its surface. The flow is steady and topologically similar at various Re . With an increase in Reynolds number, the vortex ring downstream of the sphere shifts off-axis and the flow no longer exhibits axial symmetry. Although non-axisymmetric, the flow does, however, contain a plane of symmetry and remains steady. When the Reynolds number increases further, a third transition mode occurs, at which stability of vortex ring is lost. The flow is now unsteady but periodic. The vortex is shed from the sphere surface with a single dominating frequency. A continued increase in the Reynolds number would lead to very complex flow behaviors and is beyond our investigation. These abundant phenomena and the critical Reynolds numbers at which the transition modes occur have been explored by many researchers. Taneda (1956), using the flow visualization method, identified that the generation of an axisymmetric vortex ring occurred at Re ≈ 24 . Magarvey & Bishop (1961), through the dye visualization, found that the wakes behind the liquid spheres exhibited the same vortex structure as that observed by Taneda (1956). Besides, he noticed that the stable and axisymmetric rings persisted up to Re = 210 but developed into a non-axisymmetric pattern characterized by two parallel threads in the range 217      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows of 210 < Re < 270 . This double-thread, beyond Re = 270 , lost its stability and shed from the sphere. Numerically, by employing a spectral element method, Tomboulides (1993) predicted the initial separation at Re = 20 and a transition from axisymmetric to double-thread wakes at Re = 212 . The stability of the axisymmetric flow was also examined by Natarajan & Acrivos (1993) using a finite-element method, who suggested a regular bifurcation at Re = 210 . Johnson & Patel (1999) investigated the flow regimes at Reynolds numbers up to 300 both numerically and experimentally. The lower and upper Reynolds number limits for steady axisymmetric regime was reported to be 20 and 210 while the onset of a periodic vortex shedding flow was recorded around Re = 270 . There are still many other excellent efforts on the flow characteristics investigations, including the shedding frequency measurement (Achenbach 1974; Kim & Durbin 1988; Sakamoto & Haniu 1990), vortex structure visualization (Shirayama 1992), etc. The convective heat transfer from an isolated sphere has been the subject of extensive investigations, as summarized by Clift et al. (1978) and Polyanin et al. (2002). Ahmed & Yovanovich (1994) proposed an approximate analytical solution in the range of Reynolds number ≤ Re ≤ ×104 based on linearization of the energy equation. Whitaker (1972), by examining his experimental data, provided a correlation for the average Nusselt number in a wide range of Reynolds numbers ≤ Re ≤ 105 . Numerically, Dennis et al. 218      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows (1973) calculated the heat transfer from an isothermal sphere at low values of Reynolds numbers up to 20. Dhole et al. (2006) investigated the heat transfer characteristics in the steady symmetric flow regime for both the constant temperature and constant heat flux boundary conditions on the solid sphere surface. The convective heat and mass transfer from an isolated stationary sphere is governed by two characteristic parameters: the Reynolds number Re and the Prandtl number Pr . In the current simulation, Pr is fixed at 0.71 and Re is taken as 100, 200, 250 and 300, which covers all the three flow regimes identified in the literature. For the steady axisymmetric case ( Re = 100 and 200), both kinds of thermal boundary conditions have been considered on the sphere surface, due to the availability of published results for convenient comparison. They are set as: the isothermal condition TB = and the isoflux condition − ∂T = . As for cases in the other two flow regimes ( Re = 250 ∂n and 300), calculations are carried out for the isothermal condition ( TB = ) only. A computational domain of size 25 D × 20 D × 20 D is used, with the sphere located at ( 15D , 10D , 10D ). The domain is discretized by a non-uniform mesh with a fine resolution of h = Δx = Δy = Δz = D / 40 around the sphere. For the unsteady flows, a time step size of Δt = 0.001 is selected. 219      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows 7.1.1 Steady axisymmetric flow regime In a certain low Reynolds number range, a steady axisymmetric vortex ring is reported to form behind the sphere (Taneda 1956; Tomboulides 1993; Magnaudet et al. 1995), which is often referred to as the “steady axisymmetric flow regime”. Two Reynolds numbers of Re = 100 and 200 are selected for the simulation and description of flow behaviors. The three-dimensional vortex structures identified using the λ2 -definition proposed by Jeong & Hussain (1995) are plotted in Fig. 7.1(a)-(b), where the axisymmetry is clearly visualized, being toroidal and concave toward the sphere tail (without special illustration, the three-dimensional vortex structures in the following are all identified using the method of Jeong & Hussain (1995)). Streamlines in the ( x, y) -plane at both Reynolds numbers are presented in Fig. 7.2. As can be observed, the flow separates from the surface of the sphere and then rejoins, forming a closed separation bubble which attaches to the sphere surface. It is also noted that the flows in both planes are symmetric about the centerline, and their topologies keep identical. Variations exist only in the separation location, position of the vortex center and length of the recirculation region. With Reynolds number increasing from Re = 100 to 200, the separation point on the sphere surface moves towards the front stagnation point while the vortex center extends downstream and the recirculation region becomes stretched. 220      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows The steady-state thermal fields are plotted in terms of temperature contours (isotherms) in Figs. 7.3 and 7.4, for isothermal condition and isoflux condition, respectively. As expected, the isotherms are symmetric about the centerline for both types of thermal condition. They cluster heavily around the front surface of the sphere, indicating a large heat transfer rate there. As the Reynolds number increases from 100 to 200, the clustering of isotherms on the front surface shows some enhancement. Moreover, the isotherms around the back stagnation point which is thinly distributed for Re = 100 , are more densely spaced at Re = 200 . These observations can be further verified in Fig. 7.5, where the local Nusselt number distribution along the sphere surface is presented (the angle θ is measured in the clockwise direction from the front stagnation point ( θ = 0° ) to the rear one ( θ = 180° )). Each curve manifests two peaks located around the front and rear stagnation points, with the former much higher than the latter. The Nusselt number throughout the sphere surface is enhanced as Re increases from 100 to 200. The profiles reported by Dhole et al. (2006) for both boundary conditions are included in Fig. 7.5 as well, which show a good agreement with ours. The present results are also quantitatively validated in Table 7.1 by making a comparison of our calculated drag coefficient with the numerical measurements of Johnson & Patel (1999), Gilmanov et al. (2003) and White (1974). Note that for all the problems in this chapter, the drag coefficient is 221      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows defined as CD = FD π D2 ρU ∞2 , (7.1) where FD is the drag force acting on the sphere. The comparison shows that our results in the steady axisymmetric flow regime match well with those in the literature. A further validation is implemented by examining the heat transfer rate on the sphere surface. For the isothermal case, Table 7.2 lists the surface-averaged Nusselt number Nu obtained from the present results and those calculated from the published correlations in the literature. Our numerical value is calculated by taking an area-integral of the local Nusselt number over the sphere surface and then making an average in the way Nu = 4π D ∫ Nu ids (7.2) Γ where Γ denotes the sphere surface. We can see that our results agree reasonably well with the correlations reported by Ljachowski (1940), Froszling (1938), Whitaker (1972) and Feng & Michaelides (2000) 7.1.2 Steady planar-symmetric flow regime The flow at Re = 250 is taken as a representative to illustrate the flow features in the steady planar-symmetric flow regime. Streamlines in both ( x, y) -plane and ( x, z ) -plane are presented in Fig. 7.6, which clearly reveals 222      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows the loss of the axial-symmetry in the current flow regime. Although non-axisymmetric, the flow remains steady and exhibits symmetric in the ( x, z ) -plane. It is also noted that the rear stagnation point in the ( x, y) -plane moves forward along the sphere surface and stays away from the centerline. The three-dimensional vortex structure is plotted in Fig. 7.1(c) where the vortex behind the sphere is different from the toroidal structure of axisymmetric case, and is developed into a double-threaded structure in the planar-symmetric flow regime. While the isotherms in the ( x, z ) -plane still keep symmetric about the centerline, the onset of non-axial symmetry, as expected, results in an asymmetric behavior of thermal field in the ( x, y) -plane (Fig. 7.7). The local Nusselt number along the sphere surface in the ( x, z ) -plane is depicted in Fig. 7.8. As compared to the axisymmetric cases ( Re = 100 and 200), the heat transfer is enhanced on the surface around the front stagnation point. In the ( x, y) -plane (Fig. 7.8(b)), the heat transfer rate on the front hemisphere ( ≤ θ ≤ 100 and 260 ≤ θ ≤ 360 ) is coincident with that in the ( x, z ) -plane, showing a symmetric behavior. On the rear hemisphere, anti-symmetry happens and the local peak no longer appears at θ = 180 but moves forward to θ = 170 . Additionally, the peak value increases as compared to that in the ( x, z ) -plane. 223      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows The fluid and thermal behaviors at Re = 250 , in forms of average drag coefficient and Nusselt number, are compared with those established ones in Tables 7.3 and 7.2 respectively, from which good agreements are observed. 7.1.3 Unsteady periodic flow regime As Re is increased above approximately 270, instability becomes so pronounced that flow unsteadiness is triggered. In this periodic unsteady flow regime, flow at a Reynolds number of 300 was chosen as the case of interest. The time evolution of drag coefficient and surface-averaged Nusselt number on the sphere is well traced and plotted in Fig. 7.9, from which a perfect periodic characteristic is observed, showing that the vortex is shedding periodically from the sphere. The vortex shedding frequency is frequently described by the dimensionless Strouhal number, which, based on our calculation, is 0.133 and basically agrees well with the published value of 0.136 reported by Tomboulides (1993), and 0.137 provided by Johnson & Patel (1999). The instantaneous in-plane streamlines in one vortex shedding cycle are plotted in Figs. 7.10 and 7.11, corresponding to four equal-interval phases. It is clear from the pictures that the streamlines in the ( x, z ) -plane (Fig. 7.10) remain symmetric throughout the cycle, indicating that the observed plane-symmetry for the steady flows is still present in the unsteady ones, 224      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows (b) G / D = 2.5 (a) G / D = 1.2 Fig. 7.15 Three-dimensional vortex structures for flow around a pair of tandem spheres at Re = 40 Fig. 7.16 Streamlines (left) and isotherms (right) at Re = 40 and G / D = 1.2 Fig. 7.17 Streamlines (left) and isotherms (right) at Re = 40 and G / D = 2.5 260      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows Fig. 7.18 Streamlines for an isolated sphere at Re = 40 Fig. 7.19 Local Nusselt number distribution on the sphere surface along the circumferential direction for G / D = 1.2 Fig. 7.20 Local Nusselt number distribution on the sphere surface along the circumferential direction for G / D = 2.5 261      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows        (a) G / D = 1.5 (b) G / D = 2.0   Fig. 7.21 Three-dimensional vortex structures for flow around a pair of tandem spheres at Re = 300   Fig. 7.22 Streamlines (left) and isotherms (right) at Re = 300 and G / D = 1.5 (a) (x,y)-plane (b) (x,z)-plane Fig. 7.23 Streamlines (left column) and isotherms (right colum) at Re = 300 and G / D = 2.0 262      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows Fig. 7.24 Local Nusselt number distribution on the sphere surface in ( x, z ) -plane: a comparison between G / D = 1.5 and G / D = 2.0 Fig. 7.25 Local Nusselt number distribution on the sphere surface at G / D = 2.0    (a) t / Tshed = 0.25 263        Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows (b) t / Tshed = 0.5 (c) t / Tshed = 0.75 (d) t / Tshed = Fig. 7.26 Vortex structure evolution in one vortex shedding cycle at G / D = 3.0 : perspective view (left column); side view (right column) (a) ( x, z ) -plane 264      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows (b) ( x, y) -plane Fig. 7.27 Local Nusselt number distributions on the sphere surfaces at Re = 300 and G / D = 3.0 Fig. 7.28 Instantaneous isotherms for Re = 300 and G / D = 3.0 corresponding to Fig. 7.26(a) (a) Ω = 0.3 (b) Ω = 1.0 Fig. 7.29 Three-dimensional vortex structures induced by streamwise rotating sphere for different rotating speed at Re = 100 265      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows Fig. 7.30 Inplane streamlines (left) and isotherms (right) for Re = 100 and Ω = 0.3 Fig. 7.31 Inplane streamlines (left) and isotherms (right) for Re = 100 and Ω = 1.0 Fig. 7.32 Comparison of local Nusselt number distributions on the sphere surface at Re = 100 for different rotating speed 266      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows (a) Ω = 0.1 (b) Ω = 0.3 (c) Ω = 1.0 Fig. 7.33 Three-dimensional vortex structures induced by streamwise rotating sphere for different rotating speed at Re = 250 267      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows (a) Ω = 0.1 (b) Ω = 0.3 (c) Ω = 1.0 Fig. 7.34 Time evolutions of the drag and lift coefficients on a streamwise rotating sphere at Re = 250 for different rotating speed 268      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows Fig. 7.35 Time histories of surface-averaged Nusselt number from a streamwise rotating sphere for different rotating speed at Re = 250 (a) Ω = 0.1 (b) Ω = 0.6 (c) Ω = 1.0 Fig. 7.36 Three-dimensional vortex structures induced by streamwise rotating sphere for different rotating speed at Re = 300 269      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows (a) Ω = 0.1 (b) Ω = 0.6 (c) Ω = 1.0 Fig. 7.37 Time evolutions of the drag and lift coefficients on a streamwise rotating sphere at Re = 300 for different rotating speed 270      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows Fig. 7.38 Time histories of surface-averaged Nusselt number from a streamwise rotating sphere for different rotating speed at Re = 300 271      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows Fig. 7.39 Overall performance of flow behavior and heat transfer from the rotating sphere in terms of time-mean drag coefficient and Nusselt number z Ro Ri θ e y g x Fig. 7.40 The geometric configuration of natural convection inside concentric or vertically eccentric spherical annulus 272      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows Fig. 7.42 Local Nusselt number along the inner sphere for different Rayleigh numbers (a) Ra = 1×103 (c) (b) Ra = 1×104 Ra = 1×105 Fig. 7.41 Steady-state streamlines (left half) and isotherms (right half) at different Rayleigh numbers 273      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows (a) e = − 0.625 (b) e=0 (c) e = 0.625 Fig. 7.43 Steady-state streamlines (left half) and isotherms (right half) at different vertical eccentricities Fig. 7.44 Local Nusselt number along the inner sphere at e = 274      Chapter Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows Fig. 7.45 Local Nusselt number along the inner sphere at e = − 0.625 and e = 0.625 275    [...]... hydrodynamic and thermodynamic interaction of two tandem spheres in a steady uniform flow at Re = 40 Zhu et al (1994) examined the effect of separation distance and Re on the drag forces for Re ranging from 20 to 130 Prahl et al (20 07) reported the variation of the drag and lift forces for Reynolds number of 50, 100 and 200 Yoon & Yang (20 07) , by performing a parametric study, estimated flow-induced forces... low one of 40 and a moderate one of 300 Two separation distances of G / D = 1.2 and 2.5 are simulated for Re = 40 , and three separation distances of G / D = 1.5 , 2 and 3 are considered for Re = 300 Their flow patterns, drag and lift coefficients and heat transfer performances are well analyzed A 228      Chapter 7 Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows computational... Meanwhile, all of the profiles in Fig 7. 27 maximize 234      Chapter 7 Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows around the front stagnation point of θ = 0° and present a local peak around the rear stagnation point of θ = 180° On the downstream sphere, the Nusselt number exhibits a double-hump profile and maximizes at the two humps corresponding to θ = 65° ~ 70 ° and θ... and negative eccentric arrangements and therefore is favored for natural convection 249      Chapter 7 Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows Table 7. 1 Comparison of drag coefficient CD for an isolated sphere immersed in a free stream at Re = 100 and 200 Case Johnson & Patel Gilmanov et al White (1999) (2003) (1 974 ) Re=100 1.112 1.153 1.18 1.122 Re=200 0 .79 ... steady and axisymmetric However, as compared to the toroidal vortex structure for the stationary case ( Ω = 0 ) in Fig 7. 1(a), the vortex for the rotating cases comprises of a shroud over the sphere and a threaded structure in the near wake The streamlines in Figs 7. 30 and 7. 31 demonstrate that with Ω increasing from 0.3 to 1.0, the vortex becomes elongated and 2 37     Chapter 7 Applications of Developed... rotational speed of Ω = 0.1 (Fig 7. 34(a)) and 0.3 (Fig 7. 34(b)), the two lift components CLy and CLz exhibit sinusoidal variations The magnitudes of the drag and lift, however, are constants and independent of time, indicating that the vortex structures created by the rotating sphere are in a frozen state At the larger rotational speed of Ω = 1.0 (Fig 7. 34(c)), the magnitudes of the drag and lift vary... streamlines and isotherms presented in the aforementioned forms At the concentric case of ε = 0 , a kidney-shaped vortex is formed inside the annulus (Fig 7. 43(b)), and as convection begins to take a dominant role, uprising plume is induced on the top of the hot inner sphere and interacts effectively with the cold surface of the outer sphere For the negative eccentric configuration of ε = − 0.625 , the development. .. number profiles plotted in Fig 7. 14, from which it is observed that the Nusselt number 225      Chapter 7 Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows in the ( x, z ) -plane (Fig 7. 14(a)) are symmetric about θ = 180° for all the four phases They peak at the front and rear stagnation points of θ = 0° and θ = 180° , with the former much larger than the latter Except for. .. 0 .79 Table 7. 2 Comparison of surface-averaged Nusselt number Present Nu from an isolated hot sphere immersed in a cold free stream Case Ljachowski Froszling Whitaker Feng & Present (1940) (1938) (1 972 ) Michaelides (2000) Re=100 6.1 6.92 6.61 6.98 6.81 Re=200 8.63 8.95 8 .72 9.19 9. 07 Re=250 9.64 9 .77 9.59 10.08 9 .72 Re=300 10.56 10.51 10.39 10.89 10.52 Table 7. 3 Comparison of drag coefficient CD for. .. effective region for natural 245      Chapter 7 Applications of Developed IBM Solvers to Simulate 3D Incompressible Thermal Flows convection is remarkably expanded for the positive eccentric geometry of ε = 0.625 The formed vortex (Fig 7. 43(c)), as is observed, becomes both larger and stronger The resultant isotherms, correspondingly, appear to be more clustered on the top surface of the outer sphere . terms of temperature contours (isotherms) in Figs. 7. 3 and 7. 4, for isothermal condition and isoflux condition, respectively. As expected, the isotherms are symmetric about the centerline for. Incompressible Thermal Flows 224  The fluid and thermal behaviors at Re 250 = , in forms of average drag coefficient and Nusselt number, are compared with those established ones in Tables 7. 3 and 7. 2. al. (20 07) reported the variation of the drag and lift forces for Reynolds number of 50, 100 and 200. Yoon & Yang (20 07) , by performing a parametric study, estimated flow-induced forces