Flow, deformation, and transport - Physical Processes in Earth and Environmental Sciences - page 201

y

δ

z

T u 2

T

T o

Turbulent burst

Fluid eddy

u 1

c T w

t o

Fluid
reservoir
at T o

x

w

Heated wall at T w

c = specific heat

w = 0

rate of change of momentum per unit mass is of order
t o /( u 2 - u 1 )
rate of change of internal energy per unit mass is of order
c ( T - T w )
for Prandtl number of about 1, heat flow rate is of order
c ( T - T w ) t o /( u 2 - u 1 )

Thermal boundary layer
thickness, 2 δ, temperature,
T , velocity, w .

Fig. 4.158 Development of a thermal plume generated from a heated
point source, T p .

Fig. 4.160 Visualization of Reynolds' analogy between thermal and
momentum flux.

passage will be resistance to convective motion established
by the viscous shear layer. Laminar flows at low Re , where
there is no motion normal to the boundary surface, must
transfer the excess heat entirely by conduction. They con-
sequently have very much lower heat transfer coefficients
than high Re turbulent flows, which have very thin viscous
sublayers. In such turbulent flows, once through the thin
sublayer barrier, heat is rapidly disseminated as convective
turbulence by upward-directed fluid bursts (Section 4.5)
shed off from the wall layer of turbulence (Fig. 4.160).

4.20.5 Generalities for thermal flows

Reynolds himself established the relationship between heat
flow and fluid shear stress. Known now as “ Reynolds' anal-
ogy ” this involves a comparison of the roles of kinematic
viscosity and thermal diffusivity when these two properties
of fluids have approximately similar values (Box 4.6).
Reynolds could proceed with his analogy because, as we
mentioned in Section 3.9, Maxwell had previously viewed
molecular viscosity as a diffusional momentum transport
coefficient, analogous to the transport of conductive heat
by diffusion. What is more natural than to express the ratio
of kinematic viscosity,

Fig. 4.159 The starting head vortex and the feeding axial column of
a laminar plume.

temperature than the fluid itself (Fig. 4.160). The process
is highly important in many engineering situations when
relatively cool fluids are forced through or over hotter
pipes, ducts, and plates. In natural situations we might
envision heat transfer into a cool wind forced by regional
pressure gradients to flow over a hot desert surface. In
such convection the buoyancy force is small compared to
that due to fluid inertia and thus the flow of heat has neg-
ligible effect on the flow field or the turbulence. Heat sup-
plied by conduction to the boundary of flowing fluid must
pass through the boundary layer. The major barrier to

, to thermal diffusivity, D td , as a
characteristic property of any fluid: / D td , is termed the

Prandtl number, Pr (Fig. 4.160), whose value is usually
quoted for thermal flows of particular fluids. To compare
the behavior of different fluid flows, not just the fluids
themselves, we make a more direct analogy with Re
(remember this expression is uL / ). The required thermal

equivalent to Re, uL/D td , is termed the Peclet number, Pe ,