Fluid Dynamics, Vol. 36, No. 2, 2001, pp. 187–191. Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2001, pp. 23–28. Original Russian Text Copyright 2001 by Gorelikov, Zubkov, and Morgun.
Numerical Investigation of Water Convection near a Density Extremum in a Square Cavity with a Translating Top Boundary A. V. Gorelikov, P. T. Zubkov, and D. A. Morgun Received April 10, 2000
Abstract — Cold-water mixed convection in a square cavity with a translating top boundary is studied numerically over a wide range of variation of the Grashof number. The evolution of the steady-state solutions with variation of the top-boundary velocity is investigated.
There have been many investigations of mixed convection. For example, fluids whose density is a linear function of temperature were studied in [1–2]. However, there is a number of fluids with a nonlinear temperature dependence of the density. One example of such a fluid is water, whose density has a maximum near 4◦ C. Many studies (for example, [3–6]) show that the density inversion effect considerably complicates the convective-flow patterns and, in some cases, results in the onset of several steady-state solutions under identical boundary conditions. Thus, mixed convection in a fluid with a density extremum is of interest. In this paper, we present the results of a numerical investigation of cold-water mixed convection in a square cavity with a translating top boundary. A finite-difference analog of the equations of hydrodynamics is obtained using the control volume method. The SIMPLER algorithm [7] is employed for solving the system of equations. The steady-state solutions are obtained by the time relaxation method. A preliminary verification of the numerical convergence, performed by comparing the results for 39 × 39 and 91 × 91 grids, showed that the 39 × 39 grid is quite sufficient for a qualitative analysis of the problem. A comparison of the test calculation results with the data from [1] indicates good qualitative and quantitative agreement (in particular, the mean nondimensional heat fluxes differ by not more than 0.5%). 1.
FORMULATION OF THE PROBLEM
We will consider cold-water mixed convection in a square cavity with a translating top boundary. The vertical walls of the cavity are thermally insulated while the top and bottom wall temperatures are constant: T1 = Tm − T0 and T2 = Tm + T0 , respectively (Tm = 4.0293◦ C, |T0 | ≤ Tm ). We will assume that the fluid is incompressible, the flow is two-dimensional and laminar, the thermophysical properties are constant, a Boussinesq-type approximation but with a nonlinear temperature dependence of the density applies, and the temperature variation associated with the heat release due to the friction-induced energy dissipation can be neglected. The following equation of state is used:
ρ = ρm (1 − β |T − Tm |b ), β = 9.297173 · 10−6 (◦ C)−b ,
b = 1.894816,
ρm = 999.972 kg/m3
Here, ρm is the density maximum at the temperature Tm [5]. 0015–4628/01/3602-0187$25.00 2001 Plenum Publishing Corporation
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With account for the above assumptions, in nondimensional variables the system of equations modeling the cold-water mixed convection takes the form: DU DU 1 = −∇P + ∆U + Gr|Θ|b eg , ∇U = 0, = Dτ Dτ Pr ν 3 (T2 − T1 ) , Pr = , eg = (0, 1) Gr = gβ L ν2 α uL tν T − Tm (p + ρm gLY )L2 x , P= X = , U = , τ = 2, Θ = L ν L T2 − T1 ρm ν 2 Here, X , Y are the nondimensional Cartesian coordinates, U and V are the longitudinal and transverse velocity components, P is a convective pressure component, τ is nondimensional time, Θ is the temperature, L is the cavity side, ν is the kinematic viscosity, Gr is the Grashof number, Pr is the Prandtl number, and α is the thermal diffusivity. The boundary conditions in nondimensional form are: X = 0, X = 1 :
U = V = 0,
Y =0:
U = V = 0,
Y =1:
U = Re,
Θ = 0.5 V = 0,
∂Θ =0 ∂X Θ = −0.5
Here, Re is the Reynolds number. The mean Nusselt number for the steady state is defined as follows: Nu = −
1 0
2.
∂Θ ∂Y
dX
(i = 0 or 1)
Y =i
ANALYSIS OF THE RESULTS
We performed a series of numerical experiments for Gr = 3.5 · 103 , 4 · 103 , 5 · 103 , 5.5 · 103 , 104 , 1.5 · 2 · 104 , and 3 · 104 in which the influence of the top-boundary velocity (0 ≤ Re ≤ 300) on the structure of the steady mixed convection and heat transfer in cold water at a fixed Prandtl number Pr = 11.59 was investigated. For Grashof numbers ranging between 3.5 · 103 and 5.5 · 103 and for natural convection only, four different types of steady-state solutions may exist [6]. The type of steady-state solution depends on the initial conditions, in particular, on the initial temperature distribution. We can enumerate the solution types as follows: 0 — Θ0 = 0, 1 — Θ0 = −0.5, 2 — Θ0 = X − 0.5, 3 — Θ0 = 0.5 − X . In this paper, these four types of steady-state solutions are used as the initial conditions for the first numerical experiments. Then, as the Reynolds number increases, the initial conditions are taken to coincide with the steady-state solutions obtained for the preceding (lesser) value of the Reynolds number. The zero-type solution is unstable and, for a moving boundary, rapidly collapses into the solution corresponding to the third type of boundary conditions. Figure 1 shows the dependence of the mean Nusselt number Nu for the steady state on Re for Gr = 5 · 103 (for Gr = 4.5·103 and 5.5·103 the dependence is similar). For Gr = 4.5·103 , 5·103 , 5.5·103 and for natural convection, three types of steady-state flow are stable [6] (see the schematic flow patterns a, b, c in Fig. 1 at the point Re = 0). In accordance with the numerical results for these Grashof numbers, at small values of Re each type of initial conditions corresponds to its own steady-state solution (branches 1–3 in Fig. 1), i. e. the initial conditions for Re = 0 transform continuously into the steady-state solutions for Re > 0. The heat flux corresponding to the steady-state solutions of branch 3 decreases with Re (Fig. 1). This is because the motion at the top wall is opposite in direction to the initial (when Re = 0) flow near the top boundary. Then, as Re increases, the steady solutions of branch 3 become unstable and go discontinuously over into branch 1 (schemes d and e in Fig. 1). 104 ,
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Fig. 1. Dependence Nu(Re) for Gr = 5 · 103 . Curves 1–3 correspond to the steady-state solutions of the first, second, and third types (schemes a, b, c) for Re = 0. The stream function patterns are shown in diagrams a–j, the broken line shows the isotherm corresponding to the density extremum
Up to certain values of Re, the heat fluxes corresponding to the steady-state solutions of branches 1 and 2 increase. This is because of the intensification of the flow in the upper part of the cavity, where the temperature distribution is relatively “stable” (Θ < 0). A further increase in Re results in the intensification of the flow in the upper eddy and the size of this eddy increases. As a result, the vertical dimension of the region with an “unstable” temperature distribution (Θ > 0) decreases, i. e. the isotherm Θ = 0 corresponding to the density extremum descends. This leads to a decrease in the intensity of the flow in the bottom part of the cavity whose basic mechanism is natural convection. As a result of this process, on a certain range of Re, the heat flux passing through the cavity decreases (Fig. 1). With further increase in Re, the steady-state solutions of branch 2 become unstable and go discontinuously over into branch 1 (schemes f and g in Fig. 1). As the value of Re increases further, the solutions of branch 1 also become unstable and “drop” onto branch 4. The structure of the solutions corresponding to branch 4 (scheme h in Fig. 1) is similar to that for branch 2, i. e. one main eddy in the bottom “unstable” region and one eddy in the top “stable” region of the cavity. On branch 4, the main mechanisms of heat transfer are forced convection at the top of the cavity and heat conduction at the bottom, the natural convection becoming less and less intense with increase in Re. As a result of this, the heat flux increases with Re. In all the experiments described above, the initial approximation was taken to coincide with the steadystate solution obtained for the preceding (lesser) value of Re (“forward run”). On a certain range of Re (for example, 10.3 < Re < 10.9 for Gr = 5 × 103 ), we managed to obtain an alternative type of steady-state solution by reversing the process, i.e. by taking the steady state solution obtained for the greater value of Re FLUID DYNAMICS
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Fig. 2. The dependence Nu(Re) for Gr = 4 · 103 (a) and Gr = 1.5 · 104 (b)
Fig. 3. Evolution of the dependence Nu(Re) with increase in Gr. The curves of the first (thick lines) and fourth (thin lines) types are plotted for Gr = 0 (a), 5 · 103 (b), 104 (c), 1.5 · 104 (d), 2 · 104 (e), 3 · 104 ( f ). The hysteresis region is shown by the broken lines
as the initial approximation (“reverse run”). Thus, a sort of hysteresis takes place (Fig. 1, the region of the breaks on branches 1 and 4, schemes i and j). For natural convection at Gr = 3.5 · 103 and 4 · 104 , only the second and third types of steady-state solutions are stable; accordingly, they were taken as the initial conditions for the first numerical experiments. Figure 2a shows the dependences Nu(Re) for Gr = 4 · 103 (for Gr = 3.5 · 103 the dependence is similar). In this case, the interaction of the natural and mixed convection also complicates the form of the dependence Nu(Re) for each branch. This dependence is now nonmonotonous and has a hysteresis. A simpler situation arises for Gr = 104 , 1.5 · 104 , 2 · 104 , and 3 · 104 . In this case, initially, i. e. when Re=0, only one stable type of steady state solution exists, namely, the first. For Gr = 1.5·104 , the dependence Nu(Re) is plotted in Fig. 2b. We note that in this case hysteresis also takes place. Comparing the numerical results obtained for different values of Gr and Re, we can formulate some general conclusions. In the case of pure forced convection (Gr = 0) (Fig. 3), the dependence Nu(Re) increases monotonously. In the case of natural convection, for the sequence of values of Gr considered in this study the function Nu(Gr) also increases. However, in the case of mixed convection the dependence Nu(Re) is essentially nonmonotonous, i. e. there exist well-expressed maxima and minima (Fig. 3). Thus, in some cases an increase in Re results in a reduction in the mean heat flux passing through the cavity. In particular, when FLUID DYNAMICS
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Gr = 2 · 104 and Re = 63.0 the mean heat flux through the cavity is reduced by 8.95% as compared with natural convection at the same value of Gr. The dependences Nu(Re) for different Gr are, in a certain sense, similar. This can be explained as follows. As Gr increases, the role of natural convection increases and, as a result, the regime of predominantly forced convection is attained at greater values of Re, which results in the stretching of curve 1 (Fig. 3, thick lines). The upward displacement of the curves Nu(Re) along the Nu axis is explained by the fact that for the sequence of values of Gr considered the dependence Nu(Gr) at Re = 0 is increasing. Summary. For Grashof numbers ranging between 3.5 · 103 and 4 · 104 and Reynolds numbers on the interval 0 ≤ Re ≤ 300, steady-state solutions of the problem of mixed convection in water near a density extremum in a square cavity with a translating top boundary are obtained. The problems of mixed convection in cold water have a more extensive and more complex set of steady-state solutions than the analogous problems of natural and forced convection. The interference between natural and forced convection may result in both a decrease and an increase in the mean heat flux passing through the cavity. For each value of Gr, there exist velocities (Re) for which the heat flux has local maxima. It is shown that, in the problem considered, there are ranges of Re on which the steady-state solutions have hysteresis regions. In a cavity filled with water with density near the inversion point, two regions, with stable and unstable temperature distributions, exist. In the problem considered this leads to the existence, under identical boundary conditions, of steady-state solutions that differ in structure and hysteresis takes place. In fluids with a monotonous dependence ρ (T ) the situation (for the problem considered) is simpler and more predictable. The calculations show that branching of the steady-state solutions is observed only at small Re (for example, Re < 2 for Gr = 2.5 · 103 ). With increase in Re only one type of steady-state solution remains (one basic eddy with the center located in the middle of the cavity), the dependence Nu(Re) increasing monotonously. In the three-dimensional formulation, the solution structure of many convection problems changes radically as compared with the two-dimensional formulation. This may also be partially true for the problem considered (when the role of forced convection is insignificant). If the influence of forced convection is fairly strong, it seems likely that the flow pattern will be quasi-two-dimensional and the effects revealed in this study will be present in the three-dimensional flow as well. Of course, this is only an assumption and the problem of the steady-state solutions for convection in a three-dimensional cavity with a translating top boundary requires further research. REFERENCES 1. M. K. Moallemi and K. S. Jang, “Prandtl number effects on laminar mixed convection heat transfer in a liddriven cavity,” Intern. J. Heat Mass Transfer, 35, No. 8, 1881 (1992). 2. R. Iwatsu and J. M. Hyun, “Three-dimensional driven cavity flows with a vertical temperature gradient,” Intern J. Heat Mass Transfer, 38, No. 18, 3319 (1995). 3. Wei Tong and J. N. Koster, “Density inversion effect on transient convection in a rectangular enclosure,” Intern. J. Heat Mass Transfer, 37, No. 6, 927 (1994). 4. C. J. Ho, S.P.Chiou, and C.S.Hu, “Heat transfer characteristics of a rectangular natural circulation loop containing water near its density extreme,” Intern. J. Heat Mass Transfer, 40, No. 15, 3553 (1997). 5. B. Gebhart and J. Mollendorf, “A new density relation for pure and saline water,” Deep-See Res., 24, No. 9, 831 (1977). 6. P. T. Zubkov and V. G. Klimin, “Numerical study of pure-water natural convection near the density inversion point,” Izv. RAN, Mekhanika Zhidk. Gaza, No. 4, 171 (1999). 7. S.Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York (1984). 8. S. P. Rodionov, “Natural convection of water in a three-dimensional rectangular enclosure heated from below ´ near the density inversion point,” In: Proc. 2nd Russi an Conf. Heat Mass Transfer. V3 [in Russian], MEI, Moscow (1998), pp. 136–140.
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2001