6A.2 ON THE SIMILARITIES OF THE ENGINEERING AND ATMOSPHERIC BOUNDARY LAYERS Guillermo Araya1, L. Castillo1, A. Ruiz-Columbie2, J. Schroeder3, and S. Basu4 1
Department of Mechanical Engineering, National Wind Resource Center, Texas Tech University, Lubbock, TX, 79409, USA 2
University College, Texas Tech University, Lubbock, TX, 79409, USA
3
Atmhosperic Science, Texas Tech University, Lubbock, TX, 79409, USA
4
Department of Marine, Earth, and Atmospheric Sciences, North Carolina State University, Raleigh, NC 27695, USA
ABSTRACT
species. In addition, a passive scalar is a
In this study, a comparison of different turbulent
diffusive additive that possesses no dynamical
flow
a
effect on the flow motion because of its low
spatially-developing boundary layer (SDBL) and
concentration. A comprehensive review about
the corresponding atmospheric boundary layer
passive scalars can be found in Sreenivasan
(ABL). Numerical data for the SDBL have been
(1991), Sreenivasan et al. (1977); and, more
obtained
Numerical
recently, in the study carried out by Warhaft
Simulations (DNS) at different Reynolds numbers
(2000). Furthermore, the concept of passive
and stream-wise pressure gradients. Turbulent
scalars and thermal fields is equivalent only
inflow conditions have been generated by means
under specific assumptions.
of the dynamic multi-scale approach (Araya et al.,
temperature difference in the thermal boundary
2011). On the other hand, observational data for
layer is assumed small, the buoyancy effects and
the ABL are available for the comparison. The
temperature dependence of material properties
idea is to test and validate the extension of the
are
engineering scaling laws to the real ABL. Those
temperature may be considered as a passive
scaling laws have been developed by carrying
scalar, and the momentum and the heat transfer
out a similarity analysis over the governing
equations are uncoupled. Moreover, in the past,
equations of the flow (George & Castillo 1997,
the problem of heat transfer has been principally
Wang & Castillo 2003).
investigated by means of experiments to explore
Keywords: DNS, thermal boundary layer, SDBL,
the temperature as a passive scalar. On the
ABL, stratification, inflow conditions.
other hand, if the temperature difference is
parameters
by
is
performed
performing
Direct
between
negligible.
As
a
Hence, if
consequence,
the
the
significant, buoyancy cannot be neglected and the boundary layer is considered to be under
1. INTRODUCTION Transport of passive scalars in turbulent
stratification. Furthermore, thermal stratification
flows plays a key role in many engineering
occurs when the high temperature variations
applications such as wind energy, electronic
provoke fluid separation due to buoyant forces.
cooling,
combustion
and
turbine-blade
film
cooling, as well as in atmospheric flows.
Moreover, direct simulations (DNS) have
Examples of passive scalars are temperature,
tried to shed some light on the transport
humidity, pollutants, or any other chemical
phenomena, particularly in the near-wall region
Corresponding author: G. Araya (
[email protected])
1
of
boundary
layers
where
most
of
the
thermal information in direct simulations of
experimental techniques have limitations due to
turbulent boundary layers in ZPG flows. The
the spatial resolution. Most of the DNS of passive
momentum thickness Reynolds number range
scalars known so far have been performed on
was 300 – 420. As in LWS, a single scale was
fully developed turbulent channels, in which
considered along the entire boundary layer for
periodic boundary conditions in the streamwise
the rescaling processes of the thermal field: the
direction can be applied (Kim and Moin (1989)
friction temperature, . Li et al. (2009) carried
and Kasagi and Iida (1999)). However, in the simulations
of
turbulent
spatially-developing
boundary layers, the periodic boundary condition cannot be prescribed because the flow is developing in the streamwise direction. Hence, the
appropriate
instantaneous
velocity
and
out DNS of a spatially developing turbulent boundary layer over a flat plate with the evolution of several passive scalars (Prandtl numbers = 0.2, 0.71, and 2) under both isoscalar and isoflux wall boundary conditions; the highest Re was
temperature profiles must be imposed at the
830. They focused on the behavior of the scalars
inflow for each time step. Lund et al. (1998),
in the outer region of the boundary layer, which
referred henceforth as LWS, have proposed a
was significantly different from a channel-flow
methodology to generate the inlet velocity profile
simulation. More recently, Wu & Moin (2010)
based on the solution downstream by assuming
have reported DNS of an incompressible ZPG
self-similarity of the flow in the streamwise
thermal boundary layer with isothermal wall
direction. LWS tested the proposed method by
conditions.
performing Large Eddy Simulations (LES) in zero
thickness Reynolds numbers, Re, was 80 - 1950
pressure gradient flows, and the range for the
and the molecular Prandtl number Pr = 1.
momentum thickness Reynolds number, Re ,
Furthermore, the DNS studies of turbulent
was 1530 – 2150. To the best of our understanding, there are only a few numerical simulations of spatiallyevolving thermal boundary layers in ZPG flows. Bell and Ferziger (1993) used a modified version of Spalart’s code, which considered the “fringe” concept to convert the non-periodic conditions into periodic ones (Spalart and Watmuff (1993)). In their study, direct simulations were performed in iso-scalar walls for an extent of momentum thickness Reynolds number from 300 to 700 and values of the Prandtl number equal to 0.1, 0.71, and 2.0. They concluded that the momentum and passive scalar fields at Pr = 0.71 were highly correlated in the near wall region. Kong et al. (2000) extended the rescaling-recycling method by LWS to generate time-dependant inflow
thermal
The
range
for
spatially-evolving
the
momentum
boundary
layers
mentioned so far have involved ZPG flows. To the best of our knowledge and based on extensive literature review, published data on thermal DNS of pressure gradient flows are very limited (Araya & Castillo (2012)). Speaking about stratified flows, Hattori et al. (2007) employed the inflow generation method by Lund et al. (1998) and Kong et al. (2000) to investigate the effects of buoyancy on the near-wall region of stable and unstable turbulent thermal boundary layers. Furthermore, Hattori et al. (2007) performed direct simulations in the approximate range of 1000–1250 for Re and concluded that thermal stratifications caused by the weak buoyant force significantly altered the structure of near-wall turbulence.
2
summarized as follows: i) the consideration of In this paper, an innovative Dynamic
different scaling laws in the inner and outer
Multi-scale method for generation of inflow
zones of
turbulent thermal information is presented based
assimilation of streamwise pressure gradients,
on the work carried out by Lund et al. (1998) and
such as APG, and ii) the implementation of a
Kong et al. (2000). The scaling laws are obtained
dynamic approach to compute inlet parameters
by performing a similarity analysis over the
at each time step (necessary during the rescaling
governing equations in the inner and outer
process) based on the solution downstream, this
regions of the boundary layer. The velocity
avoids the use of empirical correlation as in LWS,
scaling is based on the analysis performed by
so that more general flows can be simulated.
George and Castillo (1997) and successfully
Additionally, low and high order statistics from
implemented by Araya (2008), Araya et al.
present DNS of passive scalars are compared
(2009) and Araya et al. (2011) in turbulent
with experimental and numerical data from the
velocity boundary layers at zero and adverse and
literature.
favorable pressure gradients. The temperature
contrasted against observational data (ABL) from
scales
to
the 200m Met Tower located at Texas Tech
investigations performed by Wang and Castillo
University in order to assess the collapsing
(2003).
properties of the proposed scaling laws.
were
The
developed
principal
according
contributions
of
the
the boundary layer permits the
Finally,
the
SDBL
results
are
proposed inflow generation technique can be
2. DESCRIPTION OF THE MULTI-SCALE METHOD Figure 1 shows a schematic of the thermal
different scaling laws in the inner and outer
boundary layer with the inner and outer regions.
parts of the boundary layer and a test plane.
The recycle plane is conveniently located far
Table 1 shows the proposed scaling laws used
downstream from the inlet in order to ensure an
for the momentum and thermal boundary layers
almost zero value for the two-point correlation of
as well as the single scaling approach employed
any flow variable between these two planes.
by Lund et al. (1998) and Kong et al. (2000).
However, the selection of the recycle plane
The velocity and thermal scaling laws are
location is a trade off: the further the distance
obtained by performing a multi-scale similarity
between inlet and recycle planes, the longer the
analysis of the governing equations. Further
time for reaching equilibrium on flow parameters
details about the development of the proposed
between these two planes; as a consequence,
scaling laws and the procedure for turbulent
the transient time is penalized. The main
inflow generation can be found in Araya (2008).
improvement in this study is the utilization of two
3
Θ Θ
T (x)
Inflow
Outflow
Outer region Inner region y
Inlet plane
Recycle plane
Test plane
x z
Fig. 1: Schematic of the thermal boundary layer with different regions.
Table 1: Velocity and temperature scaling. Single scaling Parameter
Type Mean velocity
U
along x Mean velocity
V
along y Fluctuating
u’
velocity along x Fluctuating
v’
velocity along y Mean
temperature Fluctuating
'
temperature
Proposed scaling
Inner
Outer
Inner
Outer
u
u
u
U
U
U
u
u
u
u
u
u
u
Pr S t w
w
T* T
Pr S t w
w
T* T
U
d dx
U
U
d dx
The instantaneous temperature is expressed as a contribution of a mean value plus a fluctuation.
where * accounts for any upstream effects. In
Furthermore, in the inner region the mean
the outer region, a defect law is applied,
temperature follows a thermal law of the wall given as,
w inner Tsi ( x) g si ( yT , Pr,*)
outer Tso ( x ) g so ( yT , Pr,*)
(1)
(2)
Notice that Tsi (x) and Tso (x ) are the unknown
4
temperature scales for the inner and outer layers and only depend on x . Applying eqns. (1) and (2) to the inlet and recycle planes and assuming
that
g si ( yT , Pr,*)
functions
'inner inl ( yT ,inl , zT , t si ) 'Ti 'rec ( yT , inl , zT , t si ) (9) outer 'inl ( yT ,inl , zT , tso ) 'To 'rec ( yT ,inl , zT , tso ) (10)
and
g so ( yT , Pr,*) are the same at both streamwise
stations in the limit as u ,
It was found in Araya and Castillo (2012) that
Ti 'Ti and To 'To . Furthermore, a composite profile can be expressed in the entire thermal boundary layer by defining a weighted
inlinner Ti rec w,inl Ti w,rec
outer inl
(3)
average of the inner and outer profiles,
(4)
To rec (1 To )
inner inl 'inner 1 W ( yT ,inl ) inl
The T -parameters are the ratios of the mean temperature scales at the inlet and at the
outer inl
(11)
'outer W ( yT ,inl ) inl 'inl inl
recycle planes, for inner and outer zones,
where the weight function, W ( y T ) , is defined
respectively,
as,
Ti
Tsi ,inl Tsi ,rec
S S
Pr Pr
S t w S t w
t
w
inl
t
w
rec
W ( yT )
inl
(12) rec
(5) In the Lund’s method, the selected values for a and
To
Tso , rec
w T inl T* w T rec
(6)
used
performing
an
analysis
on
for
the
investigation. The ratios Ti and To , given by expressions (5) and (6), require the values of S t , w , T
Nomenclature section. Moreover, a similar is
after
were a = 4 and b 0.2 and also used in this
The definitions of all terms can be found in the
procedure
b
independent spatially evolving boundary layer * T
Tso ,inl
a ( y T b) 1 tanh(a) 1 tanh 2 (1 2b) y T b
temperature
fluctuations in the inner and outer regions,
and T* at the inlet and recycle planes. For the Stanton number at both stations (i.e., the inlet and recycle planes) the following equation is employed, which was developed by matching the inner and outer profiles in the overlap region, Wang et al. (2008),
'inner T ' si ( x) g ' si ( yT , zT , t si )
(7)
' outer T ' so ( x ) g ' so ( yT , zT , t so )
(8)
Hence, the fluctuations at inlet can be obtained
1 C T oT St Pr T C iT
A exp T ln T
T
2
(13)
from those at the recycle location, Furthermore, the mean temperature profile was prescribed during the transient period to guide
5
the thermal field and only the fluctuations were
is fixed as in the velocity field. The boundary
rescaled from the recycle plane. The imposed
layer thickness at the recycle plane and thermal
mean temperature during the transient at the
displacement thicknesses at both planes are
inlet plane was based on the composite profile
computed from the mean temperature profiles.
by Wang and Castillo (2003). Moreover, the
This procedure was observed to produce
thermal boundary layer thickness at inlet, T inl ,
accurate and stable results with a short thermal transient. and also weakly by the Reynolds number, it is
3. THE DYNAMIC APPROACH
not justified to use 1/8 in all cases. Therefore,
We concern ourselves with simulating a
this formulation for computing the friction
thermal boundary layer as shown in the
velocity ratio is limited in principle to ZPG
schematic in figure 1. Thermal turbulent inflow
flows. In order to extend the rescaling-
information is required at the inlet plane. The
recycling method to more general flows
Cartesian coordinates x, y and z denote the
besides the canonical boundary layer flows,
streamwise,
spanwise
Araya et al. (2011) introduced a dynamic
directions, respectively. The flow is divided
approach to calculate this power based on the
into inner and outer regions. Also, the
flow solution downstream,
wall-normal
and
instantaneous temperature is decomposed into a mean value and a fluctuation
' .
The basic idea of the rescaling–recycling method of LWS is to construct a timedependent velocity field at the inlet separately in the inner and outer regions, according to equation (11). For the inner region, the ratio of friction velocities, u, at the inlet to the recycle planes must be specified. While the stress may be measured near the wall at the recycle station, the stress at the inlet is unknown. In LWS. this problem was overcame by invoking
u / U ~ Re
(14)
by invoking a new plane between the inlet and recycle stations, called the test plane, as seen in figure 1. The reader is referred to Araya et al. (2011) for further details on the dynamic method for the velocity boundary layer. In this section,
we
are
briefly
describing
the
corresponding dynamic thermal approach, as implemented in Araya et al. (2012). It is assumed a power law of the thermal boundary layer thickness Reynolds number for the friction temperature, as follows;
an empirical law describing the evolution with
T
downstream Reynolds number, i.e. taking the
/ ~ Re T
ratio of friction velocities to be equal to
For scaling purposes, the ratio of the friction
(θrec/θinl)1/8, where 1/8 comes from the usual
temperatures at the inlet to the recycle planes
‘1/n’ power-law exponent of boundary layers
are needed,
and θ is the momentum thickness computed
T
the 1/8 power is greatly impacted by pressure
gradients and other effects such as possibly
Therefore, by considering the power law (15)
wall roughness and free-stream turbulence
at the test to the recycle planes, the unknown
through the mean velocity. Nevertheless, since
inl Re T rec Re T
(15)
inl rec
(16)
6
power, T, can be computed from the flow
T
mean solution:
0.8 0.6
T (instantaneous) -0.125 ( / Re
ln test / rec test
Re T
rec
(17)
(instantaneous) -0.125 -0.125 (u / U Re White 1974)
0.8 -0.125
ln Re T
Kays and Crawford 1993)
0.6 0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4 -0.6
-0.6 -0.8 43000
-0.8 44000
45000
46000
43000
timesteps
44000
45000
46000
timesteps
Fig. 2: Time variation of coefficients and T in zero pressure gradient (ZPG) flows. Figure 2 shows the time variation of the
Kays and Crawford (1993) and White (1974).
corresponding powers for the friction velocity
Thus, once the powers are computed, the
(a) and the friction temperature (b) in ZPG
friction velocity and temperature can be
flows at high Reynolds numbers. It is observed
calculated for any type of pressure gradient
that the computed online powers oscillate
imposed.
around the proposed empirical correlations by
7
upper surface of the computational box, Dirichlet 4. NUMERICAL MODEL
conditions are imposed for the streamwise
The physical domain and the corresponding
velocity and temperature; moreover, the stress-
boundary conditions are shown in fig. 3. For the
free condition is set for the vertical and spanwise
thermal field, an isothermal condition at the wall
components of the velocity. For adverse and
is
is
favorable pressure gradient cases (APG and
prescribed for the velocity components at the
FPG), the top surface is not flat as in the ZPG
wall. Periodic boundary conditions are assumed
case.
in the spanwise direction for both fields. The
prescribed on the mesh in such a way to
pressure is prescribed at the exit plane. At the
resemble a streamline.
considered.
The
no-slip
condition
Therefore,
a
desired
curvature
is
v w 0 y y
u U
ui 0
constant
Fig. 3: Schematic of the physical domain in ZPG flow computations Information about the domain dimensions,
developed at RPI is used, which considers a
mesh configuration and input parameters are
Finite
depicted by Table 2. In all cases, the boundary
(Streamline
layer thickness, inlet , at the inlet domain is fixed
stabilization. It has been shown (Whiting and
during the entire simulation. The incompressible
Jansen (2001)) to be an effective tool for
version of PHASTA code (Parallel Hierarchic
bridging a broad range of length scales in
Adaptive
turbulent (RANS, LES, DES, DNS) flows.
Stabilized
Transient
Analysis)
8
Element
approach Upwind
with
a
SUPG
Petrov-Galerkin)
Table 2: Parameters of the different cases considered without stratification.
that all thermal profiles at different streamwise 5. RESULTS AND DISCUSSION
pressure gradients collapse by using the Wang-
In this section, the numerical predictions
Castillo thermal scaling. However, the level of
obtained from the different cases considered
collapse given by the classical deficit law is very
without stratification, as seen in Table 2, are
poor. Furthermore, a key factor in the rescaling-
shown and discussed. Figure 4 illustrates the
recycling approach is the utilization of scaling
mean temperature profiles of all cases by
laws that are able to absorb external conditions
considering
such as pressure gradients or different Reynolds
the
Wang-Castillo
scaling
(a)
employed in the present investigation and the
numbers.
classical deficit coordinates (b). It is observed
__
25
Blackwell 1972 APG m=-0.2 Uo=11.1 m/s Re=1718--4533 Blackwell 1972 APG m=-0.15 Uo=10.1 m/s Re=2008--3734 Orlando 1974 APG m=-0.275 Uo=11.6 m/s Re=3061--5517 Present - DNS - ZPG Re = 308-385 Present - DNS - ZPG Re = 2600-2987 -6 Present - DNS - FPG k=1.510 Re = 700 Present - DNS - APG m = -0.17 Re = 438-633 Present - DNS - APG m = -0.22 Re = 1000-1500
4
X
X
Blackwell 1972 APG m=-0.2 Uo=11.1 m/s Re=1718--4533 Blackwell 1972 APG m=-0.15 Uo=10.1 m/s Re=2008--3734 Orlando 1974 APG m=-0.275 Uo=11.6 m/s Re=3061--5517 Present - DNS - ZPG Re = 308-385 Present - DNS - ZPG Re = 2600-2987 Present - DNS - FPG k=1.510-6 Re = 700 Present - DNS - APG m = -0.17 Re = 438-633 Present - DNS - APG m = -0.22 Re = 1000-1500
20
__
6
( - )/
( - )/[(w - )*T / T ]
8
15
10
X X
X
X XX X
X X
2
X XX X
XX
X
X X X
10-2
0
10-1
X
X
5
X X X XX X0
10
0
0
0.5
1
1.5
0
0.5
y / T a) Wang-Castillo scaling.
1
y / T b) Classical scaling.
Figure 4: Mean temperature in outer coordinates.
9
1.5
In
figure
5,
the
streamwise
velocity
when a FPG or APG is imposed. Additionally,
' fluctuations, u rms , and thermal fluctuations, ' rms ,
' the peak or “shoulder” observed in the u rms
are shown for the ZPG, Moderate FPG and
profile for the Case APG at y / 0.7, it is not
Strong APG cases. The idea is to analyze the
seen in the corresponding ' rms profile. Another
effects of the Reynolds number and pressure
important observation is given by the fact that
gradient on the velocity and thermal fluctuations.
streamwise velocity and thermal fluctuations
For ZPG flows at low and high Reynolds
show its maximum value closer to the wall as
numbers, there is a high similitude between
the Reynolds number increases when plotted in
streamwise velocity and thermal fluctuations for each
case.
Nevertheless,
this
outer coordinates.
correlation
' between u rms and ' rms completely disappears
4.5
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
10-2
10-1
Case ZPG Re = 350 Case ZPG Re = 2988 Case Strong APG Re = 1370 Case Moderate FPG Re = 700
4
' +rms
u' +rms
4.5
Case ZPG Re = 350 Case ZPG Re = 2988 Case Strong APG Re = 1370 Case Moderate FPG Re = 700
0
100
10-2
10-1
y/
100
y / T
a) Streamwise velocity fluctuations.
b) Thermal fluctuations.
Figure 5: Effects of Reynolds numbers and pressure gradient on thermal fluctuations.
The effects of pressure gradient on the temperature examined.
variance The
budget
corresponding
are
also
of
the
temperature
variance,
'2 / 2 , reads:
transport
equation
2 /2 1 ' 2 ui ' 1 2 ' 2 / 2 1 ' ' ' Ui u ' ' 0 (18) i 2 xi Pr xi 2 Pr xixi xi xi Convection
Pr oduction
Turbulent diffusion
10
Molecular diffusion
Dissipation
Figure 6 exhibits the different terms of eq.
location moves further from the wall in the
(18) close to the wall for the ZPG low Re ,
FPG flow. Furthermore, the peak production
Moderate APG and Moderate FPG cases. It
in the ZPG flow adopts an intermediate
can be appreciated that the peak value of
location between the peak for APG and FPG cases. In addition, a similar trend is observed
'2 / 2 production in the buffer layer is not
+
for the peak of the turbulent diffusion at y