Mixed (combined) convection is a combination of forced and free convections which is the general case of convection when a flow is determined simultaneously by both an outer forcing system (i.e., outer energy supply to the fluid-streamlined body system) and inner volumetric (mass) forces, viz., by the nonuniform density distribution of a fluid medium in a gravity field. The most vivid manifestation of mixed convection is the motion of the temperature stratified mass of air and water areas of the Earth that the traditionally studied in geophysics. However, mixed convection is found in the systems of much smaller scales, i.e., in many engineering devices. We shall illustrate this on the basis of some examples referring to channel flows, the most typical and common cases. On heating or cooling of channel walls, and at the small velocities of a fluid flow that are characteristic of a laminar flow, mixed convection is almost always realized. Pure forced laminar convection may be obtained only in capillaries. Studies of turbulent channel flows with substantial gravity field effects have actively developed since the 1960s after their becoming important in engineering practice by virtue of the growth of heat loads and channel dimensions in modern technological applications (thermal and nuclear power engineering, pipeline transport).
In a mathematical description of mixed convection in the equations of motion (the Navier-Stokes equations), both the term characterizing the pressure head loss dp/dx and the term characterizing mass forces ρ_{g} are retained. The simplest case allowing an analytical solution of the problem refers to the steady-state laminar flow at a distance from the tube inlet with a constant heat flux on a wall (q_{w} = const). Figure 1 shows the calculated velocity and temperature distributions both in an ascending flow, i.e., when the directions of forced and free near-wall convections coincide (case "a"), and in a descending flow in heated tubes, i.e., when the directions of forced and free near-wall convections are opposite (case "b"). The curves refer to different values of the Rayleigh number Ra_{A} = gβD^{4} (dT/dx)/ν^{2} = 4 Gr_{q}/Re = (4/Re)(gβq_{w}D^{4}/λν^{2}). In the case of an ascending fluid flow, velocity profiles peak near the wall giving an M-like shape which becomes sharper with the growth of the effect of thermogravitation convection until instability of the laminar flow occurs before the calculated values of the axis velocity have vanished (Figure 1a, Ra_{A} = 625). In a descending flow the stability is disturbed rather quickly, as indicated by the velocity profiles (Figure 1b) which even at relatively small growth of the bouyancy effect acquire zero values of the velocity gradient on the wall. A remarkable property of temperature distributions should be noted for all the cases considered (Figure 1a and b); even with strong deformation of velocity distribution, the temperature profiles only differ slightly from the profiles for pure forced convection. The shown specific features of velocity and temperature shown in Fig. 1 are reflected on the corresponding variation of the relative heat transfer value presented in the left portion of Fig. 2 where Nu_{0} means the Nusselt number for a "purely" forced flow.
The variation of heat transfer with Grq in turbulent and laminar mixed convection is considerably different (Figure 2). In the case of a descending flow (b) (Nu/Nu_{0}), falls with Gr_{q} in laminar mixed convection due to the retardation near a wall, but with turbulent mixed convection (Nu/Nu_{0}) grows as the heat load increases due to additional flow turbulization. For an ascending turbulent flow the behavior of the curve (Nu/Nu_{0}) — Gr_{q} is nonmonotonic. First, heat transfer deteriorates with increasing Gr_{q}. This is caused by the attenuation of turbulent momentum and heat transfer as a result of the effect of buoyancy forces on the shear stress profile and hence as the generation of turbulence. With further increase in the Grashof number, heat transfer begins to grow, thus reflecting an intensive development of the free convection effect on the flow as a whole as also takes place in the laminar mode (case "a"). In turbulent flow, the velocity profiles also acquire a typical M-like shape, but the quantitative characteristics, as well as the mechanism of momentum and heat transfer, are quite different for turbulent and laminar mixed convection. In the region of a strong effect of free convection the mode of thermal turbulent convection with the dependence of the Nusselt number on the Grashof number, characteristic to turbulent free convection Nu = A(Pr)Gr^{1/4} and excluding the effect of a geometry parameter on heat transfer, is established. Line B in Figure 2 is constructed by the relation for the Nusselt number with developed free convection on a vertical plate. Under normal conditions (in particular, for air at atmospheric pressure and temperatures close to room ones, the data of which are given in Figure 2) the effect of the gravity field on a forced turbulent flow manifests itself only at relatively small Reynolds numbers, of the order of 10^{4} for tubes, and with channels of rather large dimensions equal with characteristic equivalent diameters of the order of 1 m and more. However, for media with strong density variation, e.g., under near-critical conditions where density varies from the values typical of gases to the values characteristic for a liquid, mixed convection is realized in tubes of small diameters (of the order of several millimeters) and at large values of the Reynolds number (see Figure 3 for a water flow p/p_{c} = 1.1 in a diameter 3 mm tube at Re = (2−3) × 10^{4}).
Figure 2. Variation of Nusselt number Nu (= αD/λ, where α = least transfer coefficient, D = tube diameter and λ = thermal conductivity) with Gr_{q} (= gβ D^{4}/λw).
Figure 3. Temperature profiles as a function of fluid enthalphy (h_{GC}) for ascending (a) and descending (b) flow in a 3 mm tube at (p/pc) = 1.1. (T_{w} = wall temperature, G_{ct} = mean fluid temperature).
In a turbulent flow in the presence of heat transfer, the gravity field leads not only to the manifestation of large-scale free convection covering the entire flow as in the laminar mode, but also to the manifestation of local effects on turbulence that radically change the character of heat transfer (see Figure 2). One may judge the character of the manifestation of the effect of the buoyancy forces on turbulence from the velocity distributions given in Figure 4. These velocity profiles are plotted in the near-wall region in terms of the universal parameters u^{+} = u/u* and γ^{−} = u*ρy/η, where u* is the friction velocity ( ) and y^{+} the distance from the wall. The data are for a horizontal channel. The channel can either be heated from above giving stable stratification or from below giving unstable stratification. Points 1 and 2 represent the profile near the bottom wall for a case with stable density distribution (stable stratification) and from points 4 and 5 show the profile with unstable density distribution (unstable stratification); those results may be compared with the universal velocity distribution in an equilibrium wall flow (points 3). Points 2 and 4 indicate to a small effect of density stratification on turbulence. With strong stable stratification, turbulent transport of fluid particles is retarded and a turbulent flow, which occurred under isothermal conditions, laminarizes at the same Reynolds number and the velocity distribution in this case (points 1) approaches the distribution typical of a laminar flow (a dashed line). With a strong unstable stratification (points 5) the buoyancy forces additionally turbulize the flow and in the limited case the wall logarithmic law for a velocity is transformed to the "−1/3"-law, viz.: u ~ y^{−3} (line 6) found when studying the near-Earth atmospheric turbulence. The data indicate that in physical studies and mathematical simulation of mixed convection, one should simultaneously take into account both the global effect of gravitation on the flow as a whole and its direct effect on turbulence.
REFERENCES
Petukhov, B. S. and Poliakov, A. F. (1988) Heat Transfer in Turbulent Mixed Convection, Hemisphere Publishing Corp., N.Y.
Les références
- Petukhov, B. S. and Poliakov, A. F. (1988) Heat Transfer in Turbulent Mixed Convection, Hemisphere Publishing Corp., N.Y.