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10
Turbulent Flow Heat Transfer
10.1 Reynolds-Averaged Navier–Stokes (RANS) Equation
When flow transitions into turbulence, both shear stress and heat transfer
from the surface increase due to turbulent mixing. However, in a fully turbulent region, both shear stress and heat transfer slightly decrease again due to
turbulent boundary-layer thickness growing along the surface. In this section,
we discuss external and internal flow and heat transfer problem in a fully turbulence region. Figure 10.1 shows a sketch of a typical 2-D turbulent boundary
layer for heated flow over a cooled flat surface and the fairly uniform velocity
and temperature profiles across the boundary layer due to turbulent mixing.
A laminar sublayer is developed at a very-near-wall region where turbulent
mixing is damped due to viscous effect. This laminar sublayer thickness is
the major resistance for velocity and temperature changing from the freestream value to the wall. In a fully turbulent region, velocity and temperature
change with time at a given location inside the turbulent boundary layer, that
is, u(x, y, t), v(x, y, t), T(x, y, t). These time- and location-dependent behaviors
make turbulent flow and heat transfer much harder to analyze as compared
to laminar boundary-layer flow and heat transfer problem.
The following is to show how to obtain the Reynolds-averaged Navier–
Stokes (RANS) equation for a fully turbulent boundary-layer flow [1–6]. The
idea is to treat a fully turbulent flow as purely random motion superimposed
on a steady (time-averaged) mean flow. This means that the time-dependent
value (such as instantaneous velocity and temperature, etc.) equals the timeaveraged value (Reynolds-averaged value) plus the fluctuation value (due to
random motion). For example, the steady (time-averaged) x-direction velocity
over a period of time can be shown as
t
1
lim
u(t) dt
u =
t→∞ t
0
and
t
u " = lim
1
t→∞ t
u
" (t) dt ∼ = 0
0
195
�
10
Turbulent Flow Heat Transfer
10.1 Reynolds-Averaged Navier–Stokes (RANS) Equation
When flow transitions into turbulence, both shear stress and heat transfer
from the surface increase due to turbulent mixing. However, in a fully turbulent region, both shear stress and heat transfer slightly decrease again due to
turbulent boundary-layer thickness growing along the surface. In this section,
we discuss external and internal flow and heat transfer problem in a fully turbulence region. Figure 10.1 shows a sketch of a typical 2-D turbulent boundary
layer for heated flow over a cooled flat surface and the fairly uniform velocity
and temperature profiles across the boundary layer due to turbulent mixing.
A laminar sublayer is developed at a very-near-wall region where turbulent
mixing is damped due to viscous effect. This laminar sublayer thickness is
the major resistance for velocity and temperature changing from the freestream value to the wall. In a fully turbulent region, velocity and temperature
change with time at a given location inside the turbulent boundary layer, that
is, u(x, y, t), v(x, y, t), T(x, y, t). These time- and location-dependent behaviors
make turbulent flow and heat transfer much harder to analyze as compared
to laminar boundary-layer flow and heat transfer problem.
The following is to show how to obtain the Reynolds-averaged Navier–
Stokes (RANS) equation for a fully turbulent boundary-layer flow [1–6]. The
idea is to treat a fully turbulent flow as purely random motion superimposed
on a steady (time-averaged) mean flow. This means that the time-dependent
value (such as instantaneous velocity and temperature, etc.) equals the timeaveraged value (Reynolds-averaged value) plus the fluctuation value (due to
random motion). For example, the steady (time-averaged) x-direction velocity
over a period of time can be shown as
t
1
lim
u(t) dt
u =
t→∞ t
0
and
t
u " = lim
1
t→∞ t
u
" (t) dt ∼ = 0
0
195