Question 2.12: Scale Analyses (for readers who like heat transfer) Task A:......
Scale Analyses (for readers who like heat transfer)
Task A: Scale Natural Convection on a Vertical Wall
Consider steady laminar “natural convection” where, for example, air near a heated vertical wall rises against gravity. Within that thin thermal wall layer, the governing equations are (see Sketch):
(air flow) \rm\underbrace{\mathrm{u}\frac{\partial\mathrm{u}}{\partial\mathrm{x}}+v\frac{\partial\mathrm{u}}{\partial\mathbf{y}}}_{\rm {inertia}}=\underbrace{\mathrm{\nu}\frac{\partial^{2}\mathrm{u}}{\partial\mathrm{y}^{2}}}_{\rm{friction}}+\underbrace{\mathrm{g\beta }(\mathrm{T}-\mathrm{T}_{\infty })}_{\rm{buoyancy}} (E.2.12.1)
where β is the volumetric expansion coefficient, \rm(\beta \approx -\frac{1}{T} \text{for air}),\,T_\infty is the ambient air temperature far away from the heated wall and T is the actual air temperature within the thermal boundary layer. Thus, a second equation for T is necessary:
(air temperature) \underbrace{\mathrm{u}\frac{\partial\mathrm{T}}{\partial\mathrm{x}}+\rm v\frac{\partial\mathrm{T}}{\partial\mathbf{y}}}_{\rm {heat\; convection}}=\underbrace{\alpha \frac{\partial^{2}\mathrm{T}}{\partial\mathrm{y}^{2}}}_{\rm{heat\; conduction}} (E.2.12.2)
Note: Equations (E.2.12.1) and (E.2.12.2) are special cases of Eqs. (2.23b) and (2.45b), respectively.
(Continuity) \rm\frac{\partial u}{\partial x} +\frac{\partial v}{\partial y} =0
(x-momentum) \rm u\frac{\partial u}{\partial x} +v\frac{\partial u}{\partial y} =-\frac{1}{\rho} \frac{\partial p}{\partial x} +\nu\left\lgroup\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2} \right\rgroup +g_x (2.23a–c)
(y-momentum) \rm u\frac{\partial v}{\partial x} +v\frac{\partial v}{\partial y} =-\frac{1}{\rho} \frac{\partial p}{\partial y} +\nu\left\lgroup\frac{\partial^2v}{\partial x^2}+\frac{\partial^2v}{\partial y^2} \right\rgroup +g_y
\rm\frac{DT}{Dt}=\alpha \nabla^2T+S_T (2.45)
(i) Nondimensionalize Eq. (E.2.12.1), using \rm{u\rightarrow \bar u\,,\,v\rightarrow \bar v\,;\,\partial x\rightarrow \ell\,,\,\partial y\rightarrow \delta}; and \rm\Delta T=T_w-T_\infty as scales, i.e., reference parameters, where l is the wall height, δ is the (variable) boundary-layer thickness, and \rm T_w is the (high) wall temperature.
(ii) Form proportionalities between different terms in these equations, representing forces and fluxes, to generate dimensionless groups and functional dependencies for (x) and \rm{\bar u(x)}.
| Sketch for Scaling | System Sketch |
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(i) Replacing the variables u, v, x, y, and T by \rm{u=\hat{u}\bar u,\,v=\hat v\bar v,\,x=\hat x\ell,\,y=\hat y \delta} and \rm T=\hat T \Delta T in Eqs. (E.2.13.1, 2), we obtain:
\rm{\left\lgroup\frac{\bar u^2}{\ell} \right\rgroup \bar u\frac{\partial \hat u}{\partial \hat x} +\left\lgroup\frac{\bar u \bar v}{\delta} \right\rgroup\hat v \frac{\partial\hat u}{\partial \hat y} =\left\lgroup\frac{v\bar u}{\delta^2} \right\rgroup \frac{\partial^2\hat u}{\partial \hat y^2}+(g\beta \Delta T) }or, dividing through by ({\bar u^2/\ell}) we obtain:
where by definition ε < 1, Re_\ell >>1, and the Richardson number
\mathrm{Ri}={\frac{\mathrm{g\beta \Delta T}\ell^{3}}{\mathrm{\nu }^{2}\left\lgroup\frac{u\ell}{\nu } \right\rgroup^2 }}\equiv{\frac{\mathrm{Gr}}{\mathrm{Re}^{2}}}\cong{\frac{\mathrm{Grashof}\ \ \#}{\mathrm{Reynolds}\ \#\,\mathrm{squared}}} (E.2.12.4)
Clearly, we obtained a dimensionless PDE for buoyancy-driven flow plus a new dimensionless group, the Richardson number which encapsulates buoyancy, inertia and viscous forces.
(ii) Scale analysis of the first convection term in Eq. (E.2.12.2) balanced by the thermal diffusion term yields with \rm{\delta T\rightarrow \Delta T=T_w-T_\infty }:
\begin{array}{c c}{{\overline{{{\mathrm{u}}}}\ {{{\rm{\frac{\Delta T}{\rho }} }}}\sim\alpha\ \frac{\Delta\mathrm{T}}{\delta^{2}}\qquad\mathrm{or}\quad\ \overline{{{\mathrm{u}}}}\sim\frac{\alpha\ell}{\delta^{2}}}}\end{array} (E.2.12.5)
Recall from Eq. (E.2.12.1) that very near the vertical wall, where the inertia effects are negligible (i.e., u < 1, ∂u/∂x < 1, and v << 1), upward buoyancy forces are counteracted by viscous forces, via:
{\mathrm{g\beta \Delta T}}\sim{\mathrm{\nu }}\frac{\bar{\rm u}}{\delta^{2}} (E.2.12.6)
Eliminating \rm{\bar u} in Eq. (E.2.12.6) with Eq. (E.2.12.5) yields
{\nu}\,{\frac{\alpha\mathrm{l}}{\delta^{4}}}\sim{\rm g}\beta\Delta Tor
\frac{{\delta}}{\ell}\sim\left\lgroup\frac{\rm{g}\beta\Delta\rm{T}}{\alpha\nu }\,\ell^{3}\right\rgroup ^{-\frac{1}{4}}=\mathrm{Ra}_{\ell}^{-1/4} (E.2.12.7)
where Ra = GrPr is the Rayleigh number, a ratio of buoyancy to thermal/viscous forces, while Pr = ν/α is the Prandtl number, relating the fluid’s kinematic viscosity and thermal diffusivity.
Replacing the wall or plate height \ell by x, we can deduce that the
boundary layer thickness (here \delta=\delta_{\rm{th}} for Pr = 1) varies as:
\rm{\delta\sim x^{1/4}} (E.2.12.8a)
Combining (E.2.12.5) with (E.2.12.7) provides an expression for the vertical mean or reference velocity, i.e.,
\rm{{{\bar u\rightarrow\sqrt{{ x}}}}\qquad\mathrm{or}\qquad u_{ref}=\sqrt{{ g}\beta ({ T}_{w}-{ T}_{\infty })\ell}} (E.2.12.8b, c)
Task B: A Note on the Reynolds Number
The Reynolds number, being the ratio of inertial vs. viscous forces, is universally considered to be “always important”. Actually, this is not always the case. For example, for Poiseuille flow (see Example 2.10) Re = 0 in light of the definition used in Example 1.1 because the inertia term \rm{(\vec v \cdot \nabla )\vec v} is identical to zero while the fluid flow is in dynamic equilibrium between the driving force (i.e., pressure gradient) and resistance (i.e., shear stress). Thus, on a case-by-case basis, some dimensionless groups may need a reinterpretation. For steady pipe flow we could rewrite the re-definition as the ratio of flow momentum to wall resistance.
\mathrm{Re}_{\mathrm{D}}={\frac{4\dot{\rm m}}{\pi\mathrm{\mu{{\mathrm{D}}}}}}:=\rm{\frac{\dot m v}{\mu v D}} (E.2.12.9a,b)
Alternatively, we could focus on the pipe entrance region in which ({\vec{\mathrm{v}}}\cdot\nabla){\vec{\mathrm{v}}}\Rightarrow \left\lgroup\mathrm{v_{r}}{\frac{{\partial}\mathrm{v_{z}}}{{\partial{\mathrm{r}}}}}+\mathrm{v_{z}}{\frac{{\partial}\mathrm{v_{z}}}{{\partial}\mathrm{z}}}\right\rgroup is non-zero and the conversion of \rm U_{inlet} to a fully-developed \mathrm{v}_{\rm z}(\mathrm{r})=2\mathrm{v}_{\rm{av}}{\Biggl[}1-\left\lgroup{\frac{\mathrm{r}}{\mathrm{R}}}\right\rgroup^{2}{\Biggr]} takes place, where \rm{Re_D \lt 2,000\,.}