effect of pulsating inlet on the turbulent flow and heat transfer past a backward-facing step

Pergamon Int. Comm. Heat Mass Transfer, Vol. 24, No. 7, pp. 1009-1018, 1997

Copyright © 1997 Elsevier Science Ltd Printed in the USA. All rights reserved

0735-1933/97 $17.00 + .00

P I l S0735-1933(97)00086-9

E F F E C T OF PULSATING INLET ON THE TURBULENT F L O W AND H E A T TRANSFER PAST A BACKWARD-FACING STEP

Alvaro Valencia Departamento de Ingenieria Mecanica

Universidad de Chile Casilla 2777, Correo 21

Santiago, Chile

(Communicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT A numerical simulation of a two-dimensional turbulent flow with pulsating inlet conditions in a backward-facing step is presented. The standard k-e turbulence model is solved in conjunction with the Reynolds-averaged momentum and energy equations using a control volume method. It was found with amplitude of oscillation of 0.2 that the primary vortex size varied through one pulsating cycle with the acceleration of the primary flow. The local wall shear stress in the separation zone varies markedly with pulsating inlet flow but the local wall heat transfer remains relatively constant in the separation zone. © 1997 Elsevier Science Ltd

Introduction

Turbulent reattachment occurs in a variety of engineering systems. A few examples of such

systems include diffusers, airfoils at angle of attack and sudden enlargements in pipes or ducts. The

single-sided sudden expansion (backward-facing step) is an appealing test configuration for studies

of reattachment because of its simple geometry, which produces a single region of separated flow with

a well-defined separation line at a fixed two-dimensional separation location.

Thangam and Speziale, [I], show that the standard k-e model of turbulence predicts the

reattachement length within 15% when the numerics are done properly to insure full resolution of the

flowfield. This is in contrast to earlier reported results that seem to indicate that the standard k-E

model underpredicts the reattachment point by 25%, an exaggerated underprediction due to the lack

1009

1010 A. Valencia Vol. 24, No. 7

of adequate resolution in those computations. A spurious underprediction of the reattachment point

can also result from the application of outflow boundary conditions at a short distance from the step

corner.

Figure 1 depicts the important features of the backward-facing step flow with relevant

nomenclature. The turbulent flow separates at the comer and forms a recirculating region of length

Xa. The inlet boundary-layer thickness, the Reynolds number based on the upstream conditions, Re,

and the expansion ratio, E, define the flow field and the reattachement length. Westphal and Johnston,

[2], indicate that X R increases as boundary-layer thickness and the expansion ratio increase for thin

separating boundary layers.

Tw

uo ,1' "// I To ~ c u l a t i n g region H ss~

FIG. 1 Schematic diagram of flow domain

An interesting application is the observation of pulsating turbulent flows in a channel with a

backward-facing step. The question: the unsteady vortex in the separation zone due the pulsating inlet

enhances the local heat transfer?, will be accesses in this work. The study of the behavior of the

turbulent flow and wall heat transfer for these conditions has not undertaken in the literature. Ma et

al., [3], studied the laminar flow and mass transfer in a sudden expansion for high Prandtl and

Schmidt numbers. Graham and Bremhorst, [4], applied the k-~ turbulence model to a fully pulsed air

jet, the features of the pulsed jet were successfully simulated. Sobey [5], carried out experimental

tests with asymmetric and symmetric two-dimensional expansions but with laminar inlet flow. The

objective of this study consists in a numerical analysis of the turbulent flow in a channel with a

backward-facing step and with pulsating inlet velocity. Special attention are paid to how the local skin

friction coefficient and Nusselt number varied in a situation with pulsating and separated turbulent

flow.

Vol. 24, No. 7 TURBULENT FLOW PAST A BACKWARD-FACING STEP 1011

Governine eouations

The Reynolds averaged two-dimensional Navier-Stokes equations in conjunction with the eddy

viscosity concept are used to describe the incompressible unsteady flow in the computational domain.

They are given here in non-dimensional conservative form.

Continuity:

°tr+OV:o O) ax aY

Momentum X:

au+au2+a(uv) aP 1 ¢ ~ a . . . . aU..], a . . . . . au aV,,, 2 Ok a--? ax av :--~+-~tL~tu+,,,~-~) -@tu+,,;tTi+T~,j-3~ (2)

Momentum Y:

a..___v+ ~uv ) , av 2 aP+ l__[2 a ((l+v,)av)+ a ((l+v,)(av+a.~u)) ] 2 ak (3) a~ aX aY aY Re aY " aY aX ax aY 3 aY

where the turbulent viscosity v, is given by:

k2 (4) Vt=Re Cp - -

E

The turbulence kinetic energy k and its dissipation rate e are computed from the standard k-e

model of Launder and Spalding, [6] :

ae auk ark 1 . a . . . v, . ak. a . . . + , , , ) a ~ ] + , , , ( ; _ ~ aS + ax ÷ av = - ~ t @ t t ~ + ~ J - ~ + - @ t ~ , o,~ a r

(5)

ae aue+av~ 1 . a . . . + v , . a ~ . + a . . . + v , . a e . ~ ~ e ~ c = (6) - - + [ (( l ) ) tU ) +c lv t~

G denotes the production rate of k which is given by:

1 aU2 av2 au aV2 G:- .~ [2( ( - -~) + ( -~) )+( . . -~+.~) ] (7)

The energy equation is:

ao+auo+avo I . a .. I vt. oo. a .. i v,.do.. -- t (t + ) )+ (( + ) ] a~ ax ar : ~ -~ -F; ~ -~ -ff ~ V, ~-r J (s)


The standard constants are employed: C,=0.09, C~=1.44, C2=1.92, o~=l.0, a~=l.3, at=0.9. The

following dimensionless variables are defined:

X =x-- Y = Y ~= tU° U = u V = v P= P Re= U°2h 0=T0 k= k, e= e" v,' 2h 2h 2h Uo [Io pU~ v Tw U~ U~JEh v,- v

The lower case letters denote a dimensional variable and the u and v velocity components

correspond to the x- and y- directions of the Cartesian coordinate system. The terms in the equations

are nondimensionalized using the fluid density p, the hydraulic diameter of the upstream channel 2h,

and U o, the average velocity upstream of the channel, Fig.l.

The k-e turbulence model is valid only in high Reynols number regions. It is not applicable

in regions close to solid walls where viscous effects predominate over turbulence ones. In this work

the near-wall regions are handled with the wall functions. The wall function method assumes that

close to a solid wall the velocity and temperature profiles can be described by universal logarithmic

velocity and temperature profiles. It is also assumed that in this region the turbulence is in a state of

local equilibrium, [6].

The axial velocity component at the inlet is assumed to be a uniform velocity profile in a

channel and sinusoidal in time U=Uo( l +Asin(27tStx)), where A is the amplitude of the oscillation, and

St is the Strouhal number or the non-dimensional frequency of oscillation. Initially, for x=0, U=Uo

and V=0 at all locations. The flow over the backward facing step was computed using the geometry,

inlet and boundary conditions given by the experimental results by Papadopoulus and Volkan, [7],

the step height was 25.4 mm, and the expansion ratio was E=0.5. The freestream velocity upstream

of the step was Uo=15.6 m/s, which yielded a Reynolds number Re=52400, the boundary layer

displacement thickness in the experiments was 0.22 mm. The feestream turbulence intensity upstream

of the step was 0.7% and the inlet dissipation rate was set to C~ktS/0.01H. At the exit, the boundary

conditions for the dependent variables are obtained by setting the first derivatives in the axial direction

equal to zero.

The flow configuration for the numerical calculation can be geometrically described by the

following non-dimensional parameters: A=0.2, St=l.0, Re=52400, Pr=0.71, T , , /T o =2, E--0.5 and

L/H=I0, Figure l.


Numerical solution technique

The equation set consisting of the momentum, mass, energy and turbulence model equations

(1)-(8) was discretized using the control volume method, details of which can be conveniently found

in, for example, Patankar, [8]. The convection terms in the momentum equations were approximated

using a Power-Law schema. The SIMPLEC algorithm in the form given by Van Doormaal and

Raithby, [9], with a standard TDMA solver, were used.

A run of 60000 time steps of size equal to 0.001 units of nondimensional times, (321x33 grid

points), with pulsating inlet velocity takes more thanl04 CPU minutes on an IBM Power PC

workstation.

Results and discussion

To assess and provide a benchmark, the model is first applied to plane channel flow,

Reoh=105. The velocity log-law and correlations for skin friction and heat transfer are used to

benchmark the algorithm. Table I present the skin friction C F and Nusselt Number NuDb for the

channel flow with L/DH=60 where it can be assumed that the flow is fully developed and independent

of initial conditions, for different grid sizes. The geometry and inlet conditions are available in

Heyerichs and Pollard, [10]. The Nusselt number can be predicted within i.5% and the skin factor

within 7% with a fine grid.

TABLE 1 Channel flow

Grid Size C r %C F NUDh %Nuoh

Cor~lation,[10] 2401x21 243x41 485x61 485x81

0.00450 0.00561 0.00533 0.00506 0.00483

+24.7 +18.4 +i2.4 +7.3

200.60 237.04 221.56 208.04 197.61

+18.2 +i0.4 +3.7 -1.5

For the computations of the backward facing step with pulsating inlet flow a 321x32 grid has

been used. In order to study the grid independence, the case with steady inlet conditions was run with

201x21 and 401x41 grid points as well for Re=52400. The computation results show a difference of

only 1% in the location of reattachment length X R between the 401x41 and the 321x33 grids, Table

2. Since the computation time with 401x41 grids is nearly eight times that with 321x33 grids, the


computation with 401x41 grids is abandoned in favor of 321x32 grids. The time increment between

two successive time steps was set as a 0.001. The model can predict the experimental reattachement

length within 18%, [7]. The Reynolds shear stress profiles at different positions was also compared

with the experimental profiles from [7], the agreement was under 5% with the 321x33 grid.

TABLE 2 Backward-facing step

Grid

201x21 321x33 401x41

X R

8.1 Exp. [7]

6.350 6.844 6.925

Cfxl03 lower wall

0.320 0.332 0.343

Ctxl03 upper wall

4.608 4.566 4.372

Nu Nu lower wall apper wall

87.57 133.77 92.06 131.82 93.94 126.16

The time-dependent recirculation region in one oscillation period shows Fig. 2.

i - . . . ; : . ~ . N X N S , ' E{:: , , ,~ i , I : ~ K X £ : : ' I~

S t . x = 0 . 2 5

S t . r = 0 . 5 0

S t - x = 0 . 7 5

FIG. 2

Velocity vectors for pulsating flow in one cycle for a sinusoidal variation of the inlet velocity,

figure shows L/H<2.5.


As the fluidflux was decreased there was a general deceleration of the flow, expresed by an

adverse pressure gradient throughout the channel• At this stage was observed the maximum of the

length of the recirculation zone, this means an expansion of the vortex in the channel. The

recirculation zone is reduced by Stox<0.5. The primary vortex moves to the comer of the backward-

facing step at the maximal inlet velocity (Stox=0.25), and the vortex has the same size of steady

flow by Stox--0.5.

The skin friction coefficient variations along the lower and upper wall are displayed in Figs.

3 (a) and (b) respectively, at four time instants of the cycle, where Cf is evaluated at the mean

Reynolds number 52400. The instantaneous skin friction distribution varies markedly with the time.

The minimum of the skin friction coefficient at the lower channel wall increases by 30% with

pulsating inlet conditions. The length of the recirculation zone ( Cf=0 in Fig. 3 (a)) increases by 17%

by the minimum of the inlet velocity, Stox=0.75, instantaneous Re=41920. In contrast, the maximum

of the local heat transfer on the lower channel wall change less through one pulsating cycle, Fig. 4

(a).

The skin friction coefficient variations along the upper wall are similar to the variations of the

local heat transfer coefficient at the wall, Fig. 3 (b) and 4 (b). By Stox=0.25 the maximal inlet

velocity, instantaneous Re=62880, the local shear stress and the heat transfer are maximum and the

length of the recirculation zone is minimum.

i S g B

Cf "/.01[-4

LinK-4

8 ~ . - 8

-U .U -4

lSKt t

- t .ant-3J 0.0

• I " I " v " I "

m efiTa~._ ~ , ~ f , ~ S t T :O .Z6

• 4-0~4 6"1 1" =0 .60 S/ 1" =0 .76

t.O 4 .0 o,o 8.0 IO.O

x/H

| . O g - |

8 . R - a

¥ . 0 | - a

L211-$

6 1 1 S

1.111-~

I . I I 4 0 .0

S¢ •r =# . . i s .5'd 5" =0 .80

LO 4 .0 LO O.O IO,O

x/H

(a) (b)

FIG. 3

Variation of the local skin friction coefficient at the lower channel wall (a) and at the upper

channel wall (b) in one cycle of pulsating flow.


1 8 0 • I ' w ' I ' I •

1130

+ S t r : 0 , l + 60 + S l r =0 .60

SI "¢ :0 .76 50

400.0 |.lo 41.10 ll.lo ll.lo . . . .

x/u

!10 ZOO leo leo 17o |00

N U l o o 14o l:SO tso IiO IOO

~.o

St 1" =O.J$

i I i I , I i I i

I.o 4.0 LO S.O I0.0

x/H

(a) (b)

FIG. 4

Local Nusselt number at the lower (a) and at the upper channel wall (b) in one cycle.

0.0~44

0.0064

0.0048

0,1~40

O.O0~t

O , 0 0 ~ , 4

O+OOlO

O.O0~ll

0.0~00

E~t

St= t ,O

laO

Hu

i + I r i

: : : : : l o l H r wal~

S I = t . O

4O 61 63 O5 57 6O 40 61 83 68 67 59

T T

(a) (b)

FIG. 5

Average skin friction coefficient (a) and Nusselt number (b) at the upper and lower channel walls

as a function of time

The relationships among the location of zero shear stresses and maximum local heat transfer

is not clear with pulsating inlet flow. The reattachement point moves widely, whereas the maximum


heat transfer at the lower wall during one cycle is located close to the point seen with steady flow

at the mean Reynolds number.

The variation of the average skin coefficient and Nusselt number at the lower and upper

channel walls with the time show Fig. 5. The Nusselt number at the lower wall is not a function of

the time, Fig, 5 (b). The pulsatility of the inlet flow did not appear to impact the heat transfer in

separated regions as the skin friction coefficient.

Conclusions

The flow and heat transfer in a backward facing step with pulsating inlet conditions was

numerically simulated. The standard k-c turbulence model was solved in conjunction with the

Reynolds-averaged momentum and energy equations. The amplitude of oscillation was 0.2 and the

Reynolds number was set on 52400. It was found that the wall shear stress in the separation zone

varied markedly with the pulsating flow and the wall heat transfer remains relatively constant in the

separation zone. The length of the recirculation zone increases by 17 % at the minimum of the inlet

velocity.

The financial support received of CONICYT-CHILE under grant No 1950559 is gratefully

acknowledged.

Nomenclature

A Cf C., C~, DH E f G h H k L Nu

amplitude of oscillation skin friction coefficient C 2 k-e turbulence model constants hydraulic diameter, 2h expansion ratio, s/H frequency generation rate of turbulent kinetic energy upstream channel height downstream channel height dimensionless turbulent kinetic energy downstream channel length local Nusselt number, ct(x) 2h / k


P Pr Re S

St t T % Tw U

Uo V

X

Y XR

~(x) E 0 V

VI

(~k, OE

Ot

P

dimensionless pressure Prandtl number Reynolds number, U o 2h / v step height Strouhal number, f 2h / U 0 time temperature inlet fluid temperature channel wall temperature horizontal velocity component average velocity at the inlet vertical velocity component horizontal Cartesian coordinate vertical Cartesian coordinate first recirculating region length

local heat transfer coefficient dimensionless dissipation rate of turbulent kinetic energy dimensionless temperature molecular kinematic viscosity turbulent kinematic viscosity k-E turbulence model constants turbulent Prandtl number density dimensionless time

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

References

S. Thangam and C. G. Speziale, AIAA Journal, 30, 1314, (1992).

R. V. Westphal and J. P. Johnston, AIAA Journal, 22, 1727, (1984).

P. Ma, X. Li, and D. N. Ku, Int. J. Heat Mass Transfer, 37, 2723, (1994).

L. J. W. Graham and K. Bremhorst, J. of Fluids Engng. 115, 70, (1993).

I. Sobey, J. Fluids Mechanics, 151,395, (1985).

B. E. Launder and D. B. Spalding, Camp. Meth. App. Mech. Engng. 3, 269, (1974).

G. Papadopoulos and M. Volkan, Aspect ratio effects on the centerplane pressure and flow structure in a one-sided suddenly expanding duct, Separated Flows 1993 FED-Vol. 149, pp.9- 18, ASME, New York, (1993).

S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, (1980).

J. P. van Doormaal and G. D. Raithby, Numerical Heat Transfer, 7, 147, (1984).

K. Heyerichs and A. Pollard, Int. J. Heat Mass Transfer, 39, 2385, (1996).

Received April 29, 199 7

effect of pulsating inlet on the turbulent flow and heat transfer past a backward-facing step

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