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NIST Technical Note 1640

FDS Wall Flows Part I:

Straight Channels

R. McDermott

lOO NisrNational Institute of Standards and Technology • U.S. Department of Commerce

NIST Technical Note 1640

FDS Wall Flows Part I:

Straight Channels

R. McDermott

Building and Fire Research Laboratory

National Institute ofStandards and Technology

July 2009

%

*

U.S. Department of Commerce

Gary Locke, Secretary

National Institute of Standards and Technology

Patrick D. Gallagher, Deputy Director
Certain commercial entities, equipment, or materials may be identified in this document in order to describe an experimental procedure or concept adequately. Such

identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the

entities, materials, or equipment are necessarily the best available for the purpose.

National Institute of Standards and Technology Technical Note 1640

Natl. Inst. Stand. Technol. Tech. Note 1640, 18 pages (July 2009)

CODEN: NSPUE2
FDS Wall Flows Part I: Straight Channels

R. McDermott*

Building and Fire Research Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-8663, USA

Abstract

We perform calculations of 2D laminar and 3D turbulent channel flow with

periodic streamwise boundary conditions. Prom the laminar results we verify

that the Fire Dynamics Simulator (FDS) wall boundary condition is second-order

accurate. For the turbulent cases we adapt the wall stress model of Werner and

Wengle to FDS and verify the implementation by replicating the Moody diagram, a plot of friction factor versus Reynolds number for turbulent pipe flow.

1 Introduction

Wall flows are notoriously challenging for large-eddy simulation (LES) [3, 4, 5, 10, 11]. In

spite of their promise and sophistication, practical LES codes are resigned to model the wall

shear stress as opposed to resolving the dynamically important length scales near the wall.

In this work we introduce the Werner and Wengle (WW) wall model [13] into the National

Institute of Standards and Technology (NIST) Fire Dynamics Simulator (FDS) as a practical

first step in developing models for turbulent flow around complex geometry and over complex

terrain. Such models are required in order for FDS to accurately model, for example, tunnel

fires, smoke transport in complex architectures, and wildland-urban interface (WUI) fires

[1]. As a minimum requirement, a wall model should accurately reproduce the mean wall

stress for fiow in a straight channel. We verify that this is true for FDS by reproducing the

Moody chart, a plot of friction factor versus Reynolds number for pipe flow [8].

* Email: raiidall.incdermott@nist.gov

NIST Technical Note 1640 July 2, 2009
The remainder of this article is organized as follows. In Section 2 we describe the model

formulation. In Section 3 we give an overview of the WW model. Then, in Section 4, we

conduct a verification study of the wall boundary conditions for laminar and turbulent flows

in FDS. From this study we are able to draw quantitative conclusions in Section 5 about the

accuracy of the channel flow simulations. A derivation of the FDS implementation of the

WW model is given in Appendix A and the friction factor for 2D Poiseuille flow is derived

in Appendix B.

2 Formulation

Details of the FDS formulation are given in the Technical Guide [7] . Here we provide only

the salient components of the model necessary for treatment of constant density channel

flow.

The filtered continuity and momentum equations are:

dui duiUj 1 -^ +

^

+ ?Zii + ^^Ldt dxj p [dxi dxi dxj dx (2)

where r*,?* = p{uiU] — UiUj) is the subgrid-scale (sgs) stress tensor, here modeled by gradient diffusion with dynamic Smagorinsky [6] used for the eddy viscosity. In this work we

specify a constant pressure drop dp/dx in the streamwise direction to drive the flow. The

hydrodynamic pressure p is obtained from a Poisson equation which enforces (1).

When (2) is integrated over a cell adjacent to the wall in an LES it turns out that the

most difficult term to handle is the viscous stress at the wall, e.g. fxz\z=o, because the wallnormal gradient of the streamwise velocity component cannot be resolved. Note that the sgs

stress at the wall is identically zero. We have, therefore, an entirely different situation than

exists in the bulk flow at high Reynolds number where the viscous terms are negligible and

the sgs stress is of critical importance. The quality of the sgs model still influences the wall

stress, however, since other components of the sgs tensor affect the value of the near-wall
velocity and hence the resulting viscous stress determined by the wall model. In particular,

it is important that the sgs model is convergent (in the sense that the LES formulation

reduces to a DNS as the filter width becomes small) so that as the grid is refined we can

expect more accurate results from the simulation.

The model used for t^j = fxz\z=o in this work is the Werner and Wengle model [13] which

we describe in more detail below.

3 The Werner and Wengle Wall Model

An important scaling quantity in the near-wall region is the friction velocity, defined as

u* = y/r^/p. Prom the friction velocity we define the nondimensional streamwise velocity

u'^ = u/u* and nondimensional wall-normal distance z'^ = z/i, where £ = ii/{pu*). The law

of the wall is then given by [10, 12]

u+ = 2+ for 2+ < 5 (3)

w+ = 2.4 In 2+ + 5.2 for 2+>30, (4)

The region 5 < ^^ < 30, where both viscous and inertial stresses are important, is referred

to as the buffer layer. The upper range of the log law depends on the Reynolds number

[10, 14].

Werner and Wengle [13] propose a simplification to the law of the wall to eliminate the

mathematical difficulties of handling the buffer and log layers. Furthermore, WW suppose

that their simplified formula for the streamwise velocity holds instantaneously within the

LES. The WW wall profile is [11]

u+ = 2+ for 2+ < 11.81 (5)

u+ = A{z+f for z+ > 11.81, (6)

where ^ ==^ 8.3 and B = 1/7. Note that a power law has been substituted for the log law

and the viscous sublayer and the power law region are matched within the buffer region. A

comparison of the log law and the power law is shown in Figure 1. In the region 11.81 <
35

10^

Figure 1: The law of the wall. We have omitted the buffer layer since it is not considered in the

WW model. For 2"*" ^ 11.81 we have the viscous sublayer. For z^ > 11.81 we show a comparison

of the log law (4) (dashed line) and the WW power law (6) (solid line) with A = 8.3 and B = 1/7.

z'^ < 10^ the power law is a good approximation to the log law and for 2"*" > 10^ the power

law loosely exhibits wake region behavior for a flow with Re w 5 x 10^ [10, 14]. As we see

below, this functional behavior has consequences for high Re flows.

For the purposes of adapting the WW model to FDS we suppose that the first off-wall

velocity component u represents the WW profile averaged in the wall-normal direction (refer

to Figure 2). The density is taken as the average of the neighboring cell values and uniform

along the face. The WW model as implemented in FDS is then given by

2fl\u\

Az

a

pAz

+ r] pAz

B I'r

\u\

for 2+ ^ 11.81

for 2+ > 11.81,

(7)

(8)
U- •

A

\ Az

- X

'^W ^Xz\2,z=i^

Figure 2: Near-wall grid.

where

a

B 1_±B

P = 1 + B

l + B

7 =

A

2

1 + B

(9)

(10)

(11)

(12)

The parameters defined in (9)-(12) are introduced to provide a one-to-one correspondence

between this paper and the FDS code. Note that p, is the average of the molecular viscosity

from the neighboring cells. A detailed derivation of (8) is given in Appendix A.

In order to decide which formula to use for the wall stress, (7) or (8), we must know z'^,

which of course depends on t^j. As a practical matter of implementation, given that most

boundary layers in FDS are under-resolved, we first calculate r^, from (8); we then obtain

z^ — \ItwIp ; if -z"*" > 11.81, then the computed value of r^, is retained, else r^^ is taken

from (7), which actually involves no additional computation since the ghost cell value for

the velocity is prescribed for a no-slip wall by default.
4 Results

4.1 Laminar

As verification of the no-slip boundary condition and further verification of the momentum

solver in FDS, we perform a simple 2D laminar (Poiseuille) flow calculation of flow through

a straight channel. The FDS input files are stored in the repository [2] under poiseuille_*.

The height of the channel is i:r = 1 m and the length of the channel is L = 8 m. The number

of grid cells in the streamwise direction x is N^ — 8. The number of cells in the wall-normal

direction z is varied Nz = {8,16,32,64}. The fluid density is p = 1.2 kg m""^ and the

viscosity is 0.025 kg m~^ s~^. The mean pressure drop is prescribed to be dp/dx = — 1 Pa

m"^ resulting in Re/f « 160. The (Moody) friction factor /, which satisfies

Ap = f^^-pu\ (13)

is determined from the steady state mean velocity u which is output by FDS for the specified

pressure drop. The exact friction factor for this flow is /exact = 24/Re// (see Appendix B).

The firiction factor error \f — fexactl is plotted for a range of grid spacings Az = H/Nz in

Figure 3 demonstrating second-order convergence of the laminar velocity field.

4.2 Turbulent

To verify the WW wall model for turbulent flow we perform 3D LES of a square channel with

periodic boundaries in the streamwise direction and a constant and uniform mean pressure

gradient driving the flow. The calculation is nearly identical to the laminar cases of the

previous section, except here we perform 3D calculations and maintain cubic cells as we

refine the grid: we hold the ratio 8:1:1 between Nx.Ny-.Nz for all cases. The cases shown

below are identified by their grid resolution in the z direction. The velocity field is initially

at rest and develops in time to a mean steady state driven by the specified mean pressure

gradient. The presence of a steady state is the result of a balance between the streamwise

pressure drop and the integrated wall stress from the WW model. FDS outputs the planar

6
10^

10

0) 10 ou

10-H

10

-5

10 10

' dz

H»— FDS

--O(dz)

0(dz2)

10"

Figure 3: FDS exhibits second-order convergence for laminar (Poiseuille) flow in a 2D channel.

average velocity in the streamwise direction and once a steady state is reached this value

is used to compute the Reynolds number and the friction factor. Table 1 provides a case

matrix: nine cases for three values of specified pressure drop and three grid resolutions. The

nominal Reynolds number (obtained post-run) is listed along with the friction factor from

the most refined FDS case and the friction factor computed (iteratively) from the Colebrook

equation,

1 77

- -2.0 log 10 e/D 2.51 \

rT ^ Rev?/

(14)

which is a fit to the turbulent range of the Moody chart (see e.g. [9]). The parameter s/D

is the relative roughness where D is the hydraulic diameter of the pipe or channel and Re is

the Reynolds number based on D. For all the cases reported here the hydraulic diameter is

equivalent to the channel height, D = H, and the walls of the channel are smooth, i.e. e = 0. FDS input files are stored in the repository [2] as moody_*. To provide a qualitative picture

of the flow field, Figure 4 shows contours of streamwise velocity for the case dp/dx = — 1 Pa
Table 1: Case matrix and friction factor results for turbulent channel flow. The height of the first

grid cell Az is given in viscous units z'^ for each case. Additionally, the table gives the nominal

Reynolds number Re// and the FDS friction factor results compared to the Colebrook equation

(14).

dp/dx 2+ Re// /FDS / Colebrook rel. error

(Pa/m) A^, = 8 A^, = 16 A^, = 32 {N, = 32) Eq. (14) %

-0.01 190 95 47 5.9 X 10^ 0.0212 0.0202 4.8

-1. 1900 950 470 7.5 X 10^ 0.0128 0.0122 4.6

-100. 19000 9500 4700 9.8 X 10^ 0.0077 0.0081 6.0

m-i and N^ = 32.

In Figure 5 we replicate the Moody diagram for both the laminar and turbulent cases

presented in this work. The laminar cases provide the exact result for two different Reynolds

numbers and the turbulent cases are converging to the empirical values of the friction factor

for smooth pipes. It is interesting to compare the turbulent results in Figure 5 with the values

of 2+ shown in Table 1. Notice from the table that the z"*" values for the dp/dx = —100 cases

are in a range where the power law (6) deviates significantly from the log law (4) (see Figure

1) and this may explain why the results for this high Re cgise are somewhat grid sensitive.

5 Conclusions

In this work we have verified the FDS wall model for both laminar and turbulent flow

through straight channels. We have shown that for the laminar (DNS) case FDS is secondorder accurate. It is suggested elsewhere that, as a rule of thumb, 10 % accuracy is the

best that can be expected from friction factor calculations of turbulent flow [9]. We have

adapted the Werner and Wangle wall model to variable density flows (though only constant

8
Slice

U-VEL

m/s

15.0

13.5

12.0

10.5

9.00

7.50

6.00

4.50

3.00

1.50

0.00

8.0

Figure 4: LES of square channel flow with smooth walls and periodic streamwise boundaries using

dynamic Smagorinsky and the Werner Wengle wall model. For this image N^ = 32 and the mean

pressure drop is dp/dx = — 1 Pa m~^ resulting in Ren = 7.5 x 10^ and a friction factor of / = 0.0128.
o

.COc

o

Figure 5: The FDS Moody Chart. The sohd hne for Re < 2000 is the analytical result for 2D

Poiseuille flow, / — 24/Re (see Appendix B). The sohd lines for Re > 2000 (from the Colebrook

equation (14)) are for turbulent flow at various levels of relative roughness e/D shown on the right

axis. Stars are DNS results from FDS at a single grid resolution (Nz — 64) and the symbols are FDS results for 3D LES with dynamic Smagorinsky and the Werner Wengle wall model at three

grid resolutions {N^ = {8, 16, 32}).

10
density flows are tested here) for smooth walls and have shown that, when combined with

the dynamic model for the eddy viscosity, FDS is capable of reproducing friction factors for

a broad range of Reynolds numbers to within 6.0 % relative accuracy.

In future work we plan to:

(1) Incorporate roughness effects into the wall model for both constant density and stratified

flows relevant to atmospheric turbulence.

(2) Increase the accuracy of the FDS immersed boundary method for flow around complex

geometry (i.e. eliminate the requirement of "stair-stepped" geometry). In particular, the

results for the flows presented here should be invariant to a rotation of the computational

mesh relative to the streamwise direction of the flow. At present, this is not the case in

FDS.

(3) Validate the FDS convective heat transfer models against Nusselt number correlations

for Rayleigh-Benard convection.

A Derivation of the WW Model

To obtain (8) we take the first off-wall value of the streamwise velocity to be

1 f^^

u

I pn.z — u{z)dz, (15) Az Jo

and then substitute the WW profile for u{z) and integrate.

Let Zm denote the dimensional distance from wall where z'^ = 11.81. Equation (15)

11
becomes

u —— / u{z)diZ+ / u{z)dz

1 r fZm r^z —— I u u* dz + u u* dz

lAZ Uo Jzm

A2 LVO Jzm

^"'%*dz+f ^(7) w*d2

Az [Jo i

1 Ai

1 A^

70

ZpU

u* dz +

zpu* ' ^

/t^ W*d2

Jo fl Jzm

Az /p

l +B ^- I z dz

P

(16)

7 7/

We will integrate / and // separately. First, however, we must find a way to eliminate

the unknown Zm- To do this we equate (5) and (6) at the point where the viscous and power

law regions intersect, i.e. z'^ = 11.81 = 2+ = z-mpu* /jl.

z+ =

ZmpU

P'

^m. —

pu*

\Itv,Ip

(17)

We now have Zjn in terms of r^ and otherwise known values.

Integrating section 7 of (16) we find

P

rJo -zdz = '- '

2J1

r 2^^n

Tw 2 211^

^w iP/p) 'A^-B

2/2 T^ IP

pA^-B

2p

(18)

12
Integrating section // yields

Jz ^'Aif^yr-^\ ' z-^Az = ^^"- '»

Zm X/J'/ \ P

Az

{ }

+

1

l + B

/ _ _ _L- \ 1+B'

\ \/tJ~P

(

1 + 5 V/i

A

B . X ^^

fp\f^w\ 2

)

1 + B ^^i+B _ (nipy^^A—^

(^)

l + B

A fp\ fr^] ^ ^^,^s

l + B \JlJ V P

Plugging (18) and (19) back into (16) gives

{p/p) .^_

l + B

u

I pA

2p 1 + B\jij \p (^yM ' A^'+«-j^^A

I f P \ ._2_ 1 2 \pl\z

Rearranging for t^ we find

1 + 5 V^A^y 1 + 5 V /i / V^

l +B

1 + 5 / /i B

A \pAz

1 — 5 ,i+B / u,

I l\ f p \ ._2_

l + B 2/ V/0A2

pAz

'^%i±i?f^, ^

Tw ^ P

1 — 5 ,i+B / /i 2 \pAz

which corresponds to Eq. (9.46) in [11].

l+B

+

A \pAz

l + B f p

A \pAz

B 1 u

l +B

(19)

(20)

(21)

B Poiseuille Flow in 2D

We consider fully developed laminar flow in a 2D straight channel of length L and height H.

The momentum balance in the streamwise direction gives

dp d^u

dx dz"^

13

(22)
Integrating (22) over the height of the channel we find

^^^^-ifj^'-"^^- (23)

Prom (23) we find the average streamwise velocity is

1 f^

u = — u{z) dz

2/i dxJo^ ^

'^"W (24)

1 ^Ptt2

Vljidx

The (Moody) friction factor / is defined such that the pressure drop Ap is given by [9]

/^P = f^\pu\ (25)

Combining (24) with (25) and noting that ^ = -^ we find

A. = /|i. 1 Ap^,

12^ L

u, (26)

and finally

24^^J^ (27)

•' pHu Ren' ^ ^

References

[1] http://www.fire.nist.gov/wui.

[2] http : //code . google . com/p/fds-smv/source/browse/#svn/trunk/FDS/trunk.

[3] J. S. Baggett. Some modeling requirements for wall models in large eddy simulation.

Stanford Center for Turbulence Research Annual Research Briefs, 1997.

[4] J. S. Baggett. On the feasibility of merging LES with RANS for the near-wall region

of attached turbulent flows. Stanford Center for Turbulence Research Annual Research

Briefs, 1998.

14
[5] W. Cabot. Large-eddy simulations with wall models. Stanford Center for Turbulence

Research Annual Research Briefs, 1995.

[6] M. Germano, U. Piomelli, P. Moin, and W. Cabot. A dynamic subgrid-scale eddy

viscosity model. Phys. Fluids A, 3(7): 1760-1765, 1991.

[7] K.B. McGrattan, S. Hostikka, J.E. Floyd, H.R. Baum, R.G. Rehm, W.E. Mell, and

R. McDermott. Fire Dynamics Simulator (Version 5), Technical Reference Guide, Volume 1: Mathematical Model. NIST Special Publication 1018-5, National Institute of

Standards and Technology, Gaithersburg, Maryland, October 2007.

[8] L. F. Moody. Friction factors for pipe flow. Transactions of the ASME, 66, 1944.

[9] Bruce R. Munson, Donald F. Young, and Theodore H. Okiishi. Fundamentals of Fluid

Mechanics. John Wiley and Sons, 1990.

[10] Stephen B. Pope. Turbulent Flows. Cambridge, 2000.

[11] Pierre Sagaut. Large Eddy Simulation for Incompressible Flows. Springer, 2001.

[12] H. Tennekes and J. L. Lumley. A First Course in Turbulence. MIT Press, 1972.

[13] H. Werner and H. Wengle. Large-eddy simulation of turbulent flow over and around a

cube in a plate channel. In 8th Symposium on Turbulent Shear Flows, pages 155-168,

1991.

[14] M. V. Zagarola and A. J. Smits. Scaling of the mean velocity profile for turbulent pipe

flow. Phys. Rev. Lett, 78:239-242, 1997.

15

Nisr

NIST Technical Note 1640

FDS Wall Flows Part I:

Straight Channels

R. McDermott

National Institute of Standards and Technology • U.S. Department of Commerce