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2University College, Texas Tech University, Lubbock, TX, 79409, USA. 3Atmhosperic ..... Orlando 1974 APG m=-0.275 Uo=11.6 m/s Reθ=3061--5517. Present ...
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 xixi 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