A Generalized Wall Function - NASA Technical Reports Server (NTRS)

ICOMP-99-08

NASA/TMB1999-209398

0 A Generalized

Wall Function

Tsan-Hsing Shih Institute for Computational Louis Glenn

A. Povinelli, Nan-Suey Liu, Research Center, Cleveland,

J.L. Lumley Comell University,

National Space

Glenn

Mechanics

Ithaca,

Aeronautics Administration

Research

July 1999

Center

and

New

in Propulsion,

and Mark Ohio

York

Cleveland,

G. Potapczuk

Ohio

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A GENERALIZED

WALL Tsan-Hsing

ICOMP,

NASA

John H. Glenn

Louis A. Povinelli, NASA John H. Glenn

Shih

Research

Nan-Suey Research

FUNCTION

Center,

Cleveland,

OH 44142.

Liu, Mark G. Potapczuk Center, Cleveland, OH 44135.

J. L. Lumley Cornel]

University,

April

Ithaca,

N.Y.

21, 1999

ABSTRACT The asymptotic

solutions,

pipe or boundary to more complex

described

by Tennekes

and Lumley

(1972), for surface

layer at large Reynolds numbers are revisited. flows such as the flows with various pressure

rough surfaces, etc. boundary conditions

In computational fluid to bridge the near-wall

flows in a channel,

These solutions can be extended gradients, zero wall stress and

dynamics (CFD), these solutions region of turbulent flows so that

can be used as the there is no need to

have the fine grids near the wall unless the near-wall flow structures are required to resolve. These solutions are referred to as the wall functions. Furthermore, a generalized and unified law of the wall which

is valid for whole

sublayer) is analytically adverse and favorable wall function

surface

layer

(including

viscous

sublayer,

buffer

will be useful

not only in deriving

analytic

expressions

for surface

also bringing a great convenience for CFD methods to place accurate boundary location away from the wall. The extended wall functions introduced in this for complex

1 An

layer

constructed. The generalized law of the wall shows that pressure gradients on the surface flow is very significant.

flows with acceleration,

deceleration,

separation,

recirculation

and

and

inertial

the effect of both Such an unified flow properties

but

conditions at any paper can be used rough

surfaces.

INTRODUCTION asymptotic

numbers

solution

can be written

for the

inertiM

as (Millikan,

Ur

where

U is the

ur = V/_/p,

mean y is the

velocity, normal

sublayer

in a channel

or pipe

flow at large

Reynolds

1938)

ur is the distance

K

skin friction from

the wall,

velocity

defined

by the

_ and p are the viscosity

wall stress and

rw as

density

of

the fluid. _ _ 0.41and C

_ 5.0.

Eq.(1)

is also theoretically

va_d

and only valid

for a flat plate

boundary its formal

layer, but it has been applied to other wall bounded flows with some successes despite validity. For a boundary layer with an adverse pressure gradient and zero wall stress,

Tennekes

and Lumley

(1972)

derived

another --

asymptotic

= a In

solution

which

reads

+ t3.

(2)

Up

where up is defined by the adverse wall pressure gradient as up = [(v/p)[dP_/d!t]] _ 8 according to the experimental data of Stratford (1959). Eq.(2) has not attention

in computational

fluid dynamics.

Apparently,

near separation or re-attachment points because velocity, is nearly zero. On the other hand, Eq.(2) a small

or zero pressure

In this paper,

gradient

we will briefly

because

repeat

will become

erroneous

there the wall stress, hence will not be valid for boundary

up is nearly

the analyses

Eq.(1)

1/3, and a _ 5, been paid much for flows

the skin friction layer flows with

zero.

of Tennekes

and

Lumley

and introduce

a more

general asymptotic solution for the surface flow valid for both the zero or nonzero wall stress and the zero or nonzero wall pressure gradient. Therefore, the solution can be used for flows with acceleration, deceleration, separation and recirculation. The basic idea is to assume, at large Reynolds numbers, the existence of a surface layer distinct from the outer layer in a boundary layer flow. The existence of the law of the wa_ in the surface layer

and

solution

the

existence

of the

for the surface

velocity-defect

flow in the region

law in the

_ represents

the

thickness

of the

layer

will lead

Y

boundary

> 1. layer,

(3) l_ is the

length

viscosity of the fluid which will be defined later. The region in which Eq.(3) inertial sublayer. In the vicinity of the wall where y/_ is of order one, the significantly suppressed be obtain for both the two sublayers

is called

A simple model inertial sublayers

and this region is called viscous sublayer. inertial sublayer and the viscous sublayer. buffer

layer

where

to an asymptotic

where

Y 0 is mainly in the x direction, except near the separation or re-attachment point. In general, the wall stress

vw and the

or negative parameters

with respect as follows,

wall pressure

gradient

to x direction.

= Thus

defined

uc will never

or zero pressure define a viscous

become

(dP/dx)w

We may

V_P'r_[

are non-zero

define

a velocity

+ (vldPw_l/3

zero in any boundary

gradient because ur and up cannot length scale l_ = _'/uc. This viscous

)

and scale

with

the simplification

these

positive two wall

(w)

•

layer

flows with

either

zero wall stress

be zero at the same time. With uc we may length scale is usually very small comparing

with other length scales of the boundary layer, for example, the boundary the downstream length scale L. That is, ucS/_' >> 1 and u_L#, >> 1. Let us start

can be either uc using

of the governing

equations

layer

(4) -(6) for the

thickness

5 and

flows in the thin

surface layer using an order of magnitude analysis (see Tennekes and Lumley, 1972). We use uc to scale both the mean velocity U and turbulent velocities u, v. Let L be the downstream length scale and _ = _'/uc be the length scale in y direction. With cOU/cOx ,._ uc/L and COV/Oy _ V/_,, the

continuity

u_t,,/L

equation

2. The orders

(4) gives

of magnitude

V ,'. uclJL.

The

of the turbulent

' the viscous

terms

in Eq.(6)

left

terms

hand

side

in Eq.(6)

0--;-= o

of Eq.(6)

is then

of order

are

;

(8)

are of order

=o

, v-Si-j = o \--£

/ .

(9)

Because

ucLl_'

pressure

term in the surface

>> 1 or t_lL

> 1. Eq.(14) indicates that, and the wall pressure gradient.

flow situation. the wall stress gradient constant compared is constant The

flows

Therefore,

in general, the surface The relative importance

at the wall for turbulent flows under the condition

flows are affected by both the wall of these two terms depends on the

For example, for a boundary layer flow with a strong adverse pressure gradient, _-_ can become very small and even vanish. In this case, the adverse wall pressure

controls the surface flow and the total shear stress (the left hand side of Eq.(14)) is not across the surface layer. On the other hand, if the pressure gradient is zero or small to the wall stress, or nearly we want

then

constant to consider

the two terms

the

across here

wall stress the whole include

dominates surface

acceleration,

on the right hand side of Eq.(14) 4

the surface

flow and

the total

stress

layer. deceleration could

and

even

recirculation.

be of the same order

of magnitude,

or oneis largethan the other. Notethat becauseEq.(14)is linear, we factors separately by decompose multivariate asymptotic technique we write

may deal with these two U and -_-_ into two parts (Tennekes, 1968, has shown with a that this is a valid procedure). Following Tennekes and Lumley,

u = u: + v:, __-_= -(_): The first part, second

part,

represented

represented

by

U1

and

(16)

- (_):.

is associated

--(_'-V)I,

by U2 and -(_W)2,

(15)

with the wall stress

is solely related

to the pressure

rw/p only and the

gradient

(y/p)dP_/dx,

i.e,,

yOU1 = T_;

-(_):+

_

(17)

p'

c9U2

y dP_

(18)

-(_)_ + _'--5-_-y = -pdx The first part

of the flow in Eq.(17)

length u/u_.. Similarly, and one characteristic can be written as

has only one characteristic

velocity

ur and one characteristic

the second part of flow in Eq.(18) has only one characteristic length u/up. Therefor, the nondimensional form of Eq.(17)

-(_):+ o(v_/_._) = _-_

u_

-(_-_)2

u_

There are no additional the boundary conditions

(19)

0(_ryl_) _-_'

+ O(U2/up)

(_y)

_

velocity up and Eq.(18)

vdP_/dx

(20)

p _

parameters in the boundary conditions on Eq.(19) and Eq.(20) because are homogeneous (both the mean velocity and the turbulent stress are

zero at y = 0). Therefor,

the solution

of Eq.(19)

and Eq.(20)

-- = :-:t: Ur

be of the forms:

,

put

u_

must

(21)

pu._

and

up up2 Eq.(21)-Eq.(24)

are called

(23)

p__ -

the law of the wall.

p

:_ up

g2

•

(24)

2.2

The

velocity-defect

In the outer

layer

law

of a boundary

length scale. If ur is the part of the velocity-defect

at the edge of the boundary. From II theory of dimensional and we may write

layer,

the

boundary

only characteristic ((-/1 - Uo)/(r_v/pU_) Therefore, analysis,

this system

layer

thickness

_ is the

only appropriate

velocity for the first part of the flow, then the first is a function of y, _, u_ only, where U0 is the velocity

we have a system of four quantities with two dimensions. only two independent normalized quantities can be formed

as

- Uo)lu_.' 7y} =o, F {(u_ ;_/_ or

..

= =-:zF1 Ur

For the second

part

of flow (/2, we may use the above

up The

relations

(25)

3

ASYMPTOTIC

3.1

The

If the

Reynolds

and (26) are called

inertial

-

u3

to obtain

a relation

for U2

2 law.

FOR

SURFACE

LAYER

sublayer

number,

if uc$/v

sublayer,

outer

layer.

calculated

from

Re = uc_/v,

is large

may be developed. >> 1. The existence

enough,

an overlapping

layer

between

the surface

This overlapping layer is called inertial sublayer, of the inertial sublayer can be easily seen from the

>_ 104, there

-

ucy v

will exist a region

the law of the wall in the surface Following Eq.(21)

Tennekes

and

v uc_"

Eq.(25),

and

Lumley

indicates

to a logarithmic

that velocity

the both profile

ucy/v

should

to equate

> 10 s and match

the

mean

y/_

> 1), the total

r_ y dP_ - u--_= -- Jr -_ p p dx "

(36)

or

u_ Equations

(35)

-pu_

\_/

÷ ]de,,/dxl

and (36) are the asymptotic

where u¢y/_, >> 1 and relations can be used

y/_ 1, h/L