Scale Analysis of Internal Natural Convection Dr. Om
Scale Analysis of Internal Natural Convection Dr. Om Prakash Singh Asst. Prof., IIT Mandi www.omprakashsingh.com Boundary layer thickens
Boundary layer thickens Two-dimensional rectangular enclosure with isothermal sidewalls. Governing equations The equations governing the conservation of mass, momentum, and energy at every point in the cavity are
Note that in writing these equations, we modeled the fluid as Boussinesqincompressible, in other words, = constant everywhere except in the body force term of the y momentum equation, where it is replaced by [1(TT0)].TT0)].)]. Scale analysis of governing equations Immediately after t = 0)]., the fluid bordering each sidewall is motionless: This means that near the sidewall, the energy equation (TT0)].5.4) expresses a balance between thermal inertia and conduction normal to the wall,
This equality of scales follows from recognizing T, t, and T as the scales of changes in T, t, and x in eq. (TT0)].5.4) and velocity has not yet developed (TT0)].u=v=0)].) Hence, The layer T rises along the heated wall. Scale analysis of governing equations
The velocity scale of this motion v is easier to see if we first eliminate the pressure P between the two momentum equations (TT0)].5.2) and (TT0)].5.3): This new equation contains three basic groups of terms: inertia terms on the lefthand side and four viscous diffusion terms plus the buoyancy term on the righthand side. It is easy to show that the three terms that dominate each basic group are Scale analysis of governing equations In terms of representative scales, the momentum balance (TT0)].5.8) reads
The driving force in this balance is the buoyancy effect (TT0)].g T)/T, which is not zero. T, which is not zero. It is important to determine whether the buoyancy effect is balanced by friction or inertia. Dividing eq. (TT0)].5.9) through the friction scale and recalling that 2T t yields t yieldst yields Therefore, for fluids with Prandtl number of order 1 or greater, the correct momentum balance at t = 0)]. + is between buoyancy and friction,
Scale analysis of governing equations We conclude that the initial vertical velocity scale is This velocity scale is valid for fluids such as water and oils (TT0)].Pr > 1) and is marginally valid for gases (TT0)].Pr t yields 1). The heat conducted from the sidewall into the fluid layer T is no longer spent solely on thickening the layer: Part of this heat input is carried away by the layer T rising with velocity v. Thus, in the energy equation, we see a competition among three distinct
effects: Scale analysis of governing equations As t increases, the convection effect increases [v t yields t, eq. (TT0)].5.11)], while the effect of inertia decreases in importance. There comes a time tf when the energy equation expresses a balance between the heat conducted from the wall and the enthalpy carried away vertically by the buoyant layer,
At such a time, the layer thickness is where RaH is the Rayleigh number based on the enclosure height, Scale analysis of governing equations In addition to thermal layers of thickness T, f, the sidewalls develop viscous (TT0)].velocity) wall jets. The thickness of these jets v from the momentum balance (TT0)].5.7) for the region of thickness x t yields v outside the thermal layer. In this region, the buoyancy
effect is minor, and we have a balance between inertia and viscous diffusion, Hence, In the steady state, t > tf, the fluid near each sidewall is characterized by a twolayer structure: 1. a thermal boundary layer of thickness T, f and 2. a thicker wall jet v,f t yields Pr1/T, which is not zero. 2T,f. The development of this structure is shown in Fig. 5.2.
Fig. 5.2. Development of two-layer structure near the warm wall. End
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