Simulation of unsteady turbulent flows with the effect of fluctuating external velocity by using quasi-steady models

Ivolmr-HOILAND

Simulation of Unsteady Turbulent Flows with the Effect of Fluctuating External Velocity by Using Quasi-steady Models K. Rajendran*+ Department Amrita

of Physics

Institute

of Technology

Coimbatore-641015,

and Sciences

India

ABSTRACT Simulation of unsteady turbulent boundary layer flow with the effect of fluctuating external velocity by quasi-steady models is analysed. The Problem of oscillating turbulent flows with fluctuating extemal velocity and radial frequency is very often encountered in practice in a variety of applications. Three-D representation shows that Vpar decreases and Vper increases. In the esse of 7, velocity fair saturated after some specified value. “In-plane” and “out-of-plane” velocity components show that parts of velocity fluctuate in-Phase and at a Phase of 90” with respect to the outer-flow velocity. Amplitude of velocity oscillation increases and becomes constant at certain values of o. By the influence of EV, eddy viscosity resches a maximum and then becomes constant. 0 Elsevier Science Inc., 1997

1.

INTRODUCTION

Everyday life gives an intuitive knowledge of turbulente in fluids: the smoke of a cigarette or over a fire exhibits a disordered behaviour characteristic of the motion of the air which transports it. This behavior tan be encountered in a wide range of more or less complicated situations, both in the natura1 environment and in many industrial processes. Thus, when engineers attempt to predict the behaviour of turbulent Systems, they have more to recken with than the intrinsic difficulty of the turbulente Problem. *International Centre for Theoretical Physics, P.O. Box 586, 1-34100 Trieste, Italy ‘Present address: Department of Electrical Engineering, National University of Singspore, Singspore 119260.

APPLIEDMATHEMATICSAND

COMPUTATION87:95-110

0 Elsevier Science Inc., 1997

655 Avenue of the Americas, New York, NY 10010

(1997)

00963003/97/%17.00 PI1 SOO96-300$96)00221-4

96

K. RAJENDRAN

In Order to carry out design calculations, they must also tackle the Problem of describing fluid flow through complicated physical Systems such as turbine rotors, or tube bundles in heat exchangers [l]. The wind is subject to abrupt changes in direction and velocity, which may have dramatic consequences for the seafarer or hang-glider. The rapid flow of any fluid passing an obstacle or an airfoil creates turbulente in the boundary layers and develops a turbulent wake which will generally increase the drag exerted by the flow on the obstacle. Nevertheless, ss computational methods improve and Computers grow in power, there is continuous progress in treating the merely complicated sspects of the Problem. Therefore, in many engineering applications nowadays, the major Problem faced is the irreducible one of the turbulente itself. For practical application, in the field of the aerodynamics of helicopters or turbomachinaries, it is very important to take into account the effects of the unsteadiness not only on the potential flow but also on the viscous flow, especially on the boundary layer. Within the last decade, detailed analyses or numerically generated databases have revealed new structural properties of homogeneous turbulente, hitherto unsuspected [2-61. The Problem of turbulente methods according to the number of differential equations involved and description of the methods for calculations of steady turbulent boundary layers tan be found in a review article by Reynolds [7]. The value of methods bssed on closure assumptions rests upon the fact that they predict with reasonable accuracy the mean features of the flow, which has been sufficient, at least up to now, for engineering applications. The definition of unsteady turbulent flow is in itself challenging. The basic characteristic of turbulent flow is that it is unsteady. The time dependence, however, is random and due to hydrodynamic instabilities rather than externally imposed disturbances. The Problem we want to address is the response of viscous turbulent flow to boundary conditions that Change with time, and mainly the response of turbulent boundsry layers to externally imposed transient or periodic changes. Two forms of unsteadiness are therefore involved: the erratic unsteady motion due to hydrodynanmic instability; and the organised motion due to the external disturbances, namely the fluctuating velocity, which is to be considered in this present Paper. The Problem of oscillating turbulent flows is very often encountered in practice in a variety of applications. In fact it seems to be the rule rather than the exception that flows in engineering applications are unsteady and turbulent rather than steady and laminar. Typical examples are the flows over helicopter blades or through turbomachinary cascades. For all practical purposes the analysis of such Problems is based today on quasi-steady

Simulation of Unsteady Turbulent Flow

97

models. However, it hss been experimentally demonstrated that the unsteady effects are very significant. The use of quasi-steady models is therefore not well justified. In the last few years investigators attempted to calculate unsteady turbulent Felds by integrating the complete unsteady flow equations and using quasi-steady models for their closure [SI. Although such methods are important, their success is limited. Proper and direct modeling of the specific unsteady Problem seems to be necessary. Therefore, the author would like to make an attempt to simulate the above Problem with the effect of fluctuating external velocity. External velocity is a very ambient resson for the turbulente Problem. The Problem however is challenging and of great engineering significance. The studies include the following: (i) Parallel (Vpar) and Perpendicular (Vper) velocity to the wall with the effect of external fluctuating velocity (EV) and radial frequency (w) and “in-Phase” and “out-of-Phase” velocity, Vpar and Vper with respect to time; (ii) Eddy viscosity by the influence of (w) and EV Shows that it increases, reaches a msximum and constant; (iii) Momentum thickness (0>, displacement thickness (61, and skin-friction co-efficient (C) with respect to grid Parameter for different values of EV.

2.

QUASI-STEADY

MODEL

In an effort to derive quickly engineering information badly needed in practical applications, some investigators employed straightforward extensions of existing programs to unsteady turbulent boundary-layer flows. For a two-dimensional incompressible boundary-layer flow, the continuity and momentum equations tan be written as: du

x+-=o du du du ~+U~+V-=~+U,~+Y JY

du JY

au,

au,

a*u

-+__ dY2

1 dr P JY

(2)

where U and V are the averaged velocity components that contain the mean and the organised oscillation of the flow. Some methods are bssed on algebraic relations for closing the equation, whereas others are based on one or two equation models, employing one or more differential equations and expressing the Reynolds stress in terms of the turbulent energy [9]. These are bssed on steady models. Unsteadiness in the closure sssumption enters only parametrically, via the time dependence

K. RAJENDRAN

98

of functions like the outer-flow velocity distribution or through acceleration terms in the expression for the pressure gradient or turbulente convection. In a sense, such methods tan therefore be termed quasi-steady methods because they are bssed on the steady turbulente models and the arbitrary constant and functions determined by extensive comparison with steady turbulent boundary-layer flow. Models bssed on the mixing-length concept have proved to be very successful up to now despite their simplicity. This hss been verified by Burggraff [lO] who undertook a comparative study of representative methods. Cebeci and Keller [ll] and Abbott and Cebeci [12] suggested an extension of their well tested steady-flow model to unsteady flows. They proposed to retain the expression for the inner and outer eddy viscosity models. Dynamit effects are then introduced in the turbulente models through the unsteady term in the pressure gradient as well s.s parametrically, via the instantaneous values of the displacement thickness S( 2, t> and outer-flow velocity U( z, t>. The Reynolds stress is assumed to be proportional to the mean velocity gradient

du 7= EJY

(3)

where E is a small dimensionless Parameter. A two-layer eddy viscosity model is then introduced. In the inner layer, the eddy viscosity is proportional to the velocity gradient,

du

Ei

=

p12-

dY

(4)

where p is the density and 1 is the mixing length given by

l=Ky[l-exP(-f)]

(5)

where K is a constant and A is a Van Driest damping factor. To account for flows with pressure gradients as well as flows with heat transfer, Cebeci [13] assumed that a characteristic velocity in the stokes flow

Simulation of UnsteadyTurbulent Flow

99

that models the inner part of the turbulent layer is the friction velocity at the edge of the viscous sublayer. The damping factor then becomes

where T~ is the shear stress; this assumption permits a straightforward extension to unsteady flows. The acceleration effect of the outer flow now enters via the pressure gradient

---

1 dp P dz

au, =dt+U,x,

d f4

Reynolds number based on the momentum thickness 8, is

(8) Telionis and Tsahalis [14] adopted the same inner model for the eddy viscosity

where K, is the constant, S is the displacement thickness, and Y is the intermittency factor. Cebeci and Keller [lll have limited their calculation to spatially one dimensional flows and compared their results with those of Bradshaw [15]. Dwyer et al. [lS] also developed a technique based on a quasi-steady model of the mixing-length type and integrated the boundary-layer equation by a finite-different method. he quasi-stationary mixing-length model was also used by Gupta and Trimpi [17], who computed the development of a compressible turbulent boundary layer on a semi-infinite flat plate after the passage of a shock wave and a trailing driver-gas, driven-gas interface. Cebeci [18] employed their model to calculate oscillating flows over a flat plate and compare with experimental data. The models described here may be adequate for better engineering estimates.

K. RAJENDRAN

100

3.

UNSTEADY

MODEL

Unsteadiness is taken into account either implicitly via the time dependence of the mean field, or through the d/dt terms in the model equation for turbulent energy and dissipation. Any quasi-steady model is equivalent to the assumption that the total Reynolds stress is decomposed into a mean part and an oscillating part. It is weh known that the boundary layer, whether laminar or turbulent, responds to local disturbances in an almost inviscid manner. The hypothesis here is that the outer-flow pressure fluctuations are instantly carried across the turbulent boundary layer without interaction with random fluctuations. The oscillating part of the Reynolds stress is given in terms of an eddy viscosity. du

ri=%

(10)

where E, is the eddy viscosity in the inner region. A quasi-steady version of this model yields the following expression for the Reynolds Stresses du,

ro=p12

-

i JY

2

1

du0 d%

r1 = 2p12--7 dY

dY

(11) (12)

Any quasi-steady model therefore is equivalent to the sssumption that the total Reynolds stress is decomposed into a mean part r0 and an oscillating part ri in a way determined by the governing equation. Physically it implies that the oscillating Reynolds stress r1 is proportional to the gradient of the organised oscillations d u/a y, but the eddy viscosity of the oscillating motion is proportional to the gradient of the mean flow. The differente between this model and the quasi-steady approach is that the eddy viscosity in the inner layer is assumed to be proportional to the gradient of the oscillating velocity Profile rather than the mean Profile. Purely oscillating turbulent flows would be generated if the mean part of the outer flow were Zero, i.e., U, = 0. There is no doubt that the Problem of an unsteady turbulent boundary layer is far from being solved. The emphasis in this Paper is on the computational aspects of the Problem and the physical conclusions that one

Simulation of UnsteadyTurbulentFlow

101

may draw from them. Quantities usually calculated in turbulent boundarylayer calculations are the displacement thickness 6, momentum thickness 8, local skin-friction coefficient (C), and the Reynolds number based on them and the velocities parallel and perpendicular to the Wall. In particular, outer flows were Chosen according to the formula U=

U,(l + nsin Wt)

(13)

where w is the radial frequency and q is the boundary layer thickness. An alternative approach to the Problem is based on the assumption that the disturbance is average throughout the turbulent boundary layer. However, in practice a similar hypothesis is necessary in any case in Order to discard the higher harmonics of the organised motion and their mutual interaction. Consider an outer-flow velocity distribution of the form [19].

Ue(2, t) = U,( LT)+ ;

qey

(14)

The displacement thickness (S), momentum thickness ( f?), and local skin friction co-efficient (C) are given by

6 = /,(l

- u/v,)

dy

0

CF = 2rJpu2,.

(15)

(17)

We assume the response of the viscous layer in the form

~(2, y,t> = u,(x, y) +

W(2,

y,

;[q
t) = wo(z, y) + : [

(18)

q( x, Y) ezwt + CC]+ O(?? “) (19)

where CC is the complex conjugate and u and w are the velocity components parallel and perpendicular to the wall respectively.

102

K. RAJENDRAN

Vper

15 10 5 0 -5 -10

lQG.1. Effect of Parallel(VP= cm/sec) and perpendicular (Vper cm/sec) velocities by the influenCe of fluctuating extenal velocity (EV cm/sec), with varied o as follows: (a) ,,, = 1.0 =&‘=; (b) OJ= 2.0; 63 o = 3.0; (d) o = 4.0.

Simulation

of Unsteady

Turbulent

Flow

103

1

-1

60

6

10

FIG. 2. Effect of parallel and perpendicular velocities by the influence of fluctuating extemal velocity (EV), with varied 77 as follows: (a) 1) = 1.0; (b) 7) = 4.0; (c) 1) = 8.0; (d) 9 = 10.0.

K. FtAJENDFtAN

104

We further decompose the velocity components as follows: u, = üz + 1-I,

(20)

v, = 5, + di

(21)

where bam and primes denote the ensemble average and the random fluctuation, respectively, i = 0, 1,2.. . . For this Simulation of unsteady flow, use an explicit second-Order accurate, finite-differente scheme [20]. 4.

RESULTS

AND DISCUSSION

Fig. la-d (3D) sh ows the effect of parallel and perpendicular velocities to the wall (Vpar and Vper) by the influence of fluctuating external velocity. It

Frc. 3.

Imphase and out-of-Phase velocity components with respect to grid Parameter with

varied 11BS follows: (a) TJ= 2.0 and (b) 1) = 4.0.

Simulation of Unsteady

Turbulent

Flow

105

seems that Vpar decreases and Vper increases. The same effect is observed for the range of values of radial frequency w rad/sec. The effect is maximum at the value of w = 2.0 rad/sec. From Ca>to (d), the effect of velocity initially increases by changing the value of o and then suddenly decreases at w = 3.0 and then again increases at w = 4.0. It seems that the effect of w makes the Vpar and Vper fluctuate from minimum to maximum. Fig. 2a-d Shows the same results for the various values of boundary-layer thickness (q). From this we note that there is not much Change in Vpar and Vper by

30 ,

I

25

-5 ’ 0

I 5

..

I

1

I

1

1

I

I

10

15

20

25

30

35

40

I 45

t 35 30

eo.5 1.0 1.5 2.0 2.5 3.0

0

25

0

20 vper

(b)

0

0

15 +

0 + 0 x

A *

0

9 40

10 5 0 -5 0

5

10

15

20

25

30

35

40

t FIG. 4.

Velocities parallel and perpendicular to the wall with respect to time.

45

K. RAJENDRAN

106

the effect of 17. It seems that velocity pair initially increases and gets saturated after 77= 8.0 ((c) and (d)). By the given ranges of 71,there is not much fluctuation of Vpar and Vper. In Fig. 3a-b, we plot the “in-Phase” and “out-of-Phase” velocity compo nents with respect to the grid Parameter for two different values of 7. There are the Parts of velocity fluctuation in-Phase and at a Phase of 90” with respect to the outer-flow velocity respectively. The “in-Phase” and “out-ofPhase” velocity components tan then be obtained as follows:

r&(

y) = F(

5,

45

y)cos 4 =

2,

ycos

ot

(22)

(cos2Wty2

0.3 )

I

I

1

I

w=O.5 rad/sec

0.25

I +

0.2

x

0.15 0.1 0.05 0 -0.05 0

5

10

15

20

25

30

35

40

45

J

(b)

0

5

10

15

20

EV = 0.5

25

30

35

40

45

J FIG. 5. Effect of eddy viscosity (h) with respect to grid parameter: (a) o = 0.5,1.0,1.5 and (b) EV = 0.5,1.0,1.5,2.0,2.5.

107

Simulation of Unsteady Turbulent Flow ycos

uout( 2, y) = F(

5,

y)sin 4 =

~6

(at+;1

(23)

(cos20t)1’2

where F is the amplitude function and 4 is the phsse angle. It appears that until certain ranges, almost the entire boundary layer oscillations arc in Phase with both the inner and outer flow, except for a thin

8

,

I (a)

I

7-

1

I

I ++

65-

+

+

00°

+ + +

+

I

I

+ i%=gii$ 6.0 +

-

++

+ 3

I

I

0000000-

O0 O0

2

-

1 0

I

I

12

14

16

18

20

22

12

14

16

18

20

22

I

0

0

2

4

6

8

2

4

6

8

10

I

J

10 J

FIG. 6. Velocities parallel and petpendicular to the wall with respect to grid Parameter.

108

K. RAJENDRAN

region next to the Wall. This is also the esse for oscillating laminar boundary layers. The dependence of the maximum overshoot on the boundary-layer thickness q, the velocity overshoot indicates a clearly pronounced maximum. Fig. 4a-b Shows the velocities parallel and perpendicular to the Wall with respect to time (t) for different values of radial frequency CO.It seems that velocity (Vpar) initially increases to the maximum and then decreases. The

0.008

I

I

I (a>

0.007

I

EVA.0

7

fl

2

3

4

I

2

3

4

J

5

6

i

I

1

J

5

6

i

6

J

0.7 EVd.0 3.0 . 6.0 -

0.6 0.5

C

0.4

-

-_

0.3 0.2

... .. ..... I I

0.1 0

2

I

l

3

4

5

6

7

J FIG. 7. Variation of momentum thickness (01, displacement thickness (6) and skin-friction ceefficient (C) with respect to grid Parameter.

Simulation of Unsteady Turbulent Flow

109

amplitude of oscillation of velocity decreases with the increase of o and it is nearly constant at o = 1.5 rad/sec, and then slightly increases with the increase of w. The Pattern is exactly opposite in the case of Vper: it suddenly decreases within a short time and then becomes constant. This is true for all the values of w. Fig. 5a-b Shows the effect of eddy viscosity (h) with respect to the grid Parameter for different values of (a) w and (b) external velocity (EV). Because eddy viscosity is proportional to the velocity gradient, it seems that there is no Change in the trend by varying the value of o from 0.5 to 2.0 rad/sec. But in the case of EV, eddy viscosity initially increases, reaches a maximum and decreases, and then becomes constant. The amplitude of the maximum value increases with the increase of EV. Velocities parallel and perpendicular to the Wall with respect to the grid Parameter for different values of EV are given in Fig. 6a-b. Vpar initially increases, reaches a maximum, and then becomes steady for a certain range of J. This happened for all the given values of EV. The effect is opposite in the case of Vper: it decreases and then becomes nearer to zero for all the cases of EV. Fig. 7a-c Shows the momentum thickness (01, displacement thickness (6), and skin-friction co-efficient (C) with respect to the grid Parameter for different values of EV. It Shows that both 8 and S increase with respect to J for all the given values of EV, but C decreases. REFERENCES 1 2

3 4 5 6 7 8 9 10

W. N. McComb, The Physics of Fluid Turbulenct; Clarendon Press, Oxford, 1990. W. T. Ashurst, A. R. Kerstein, R. M. Kerr and C. H. Gibson, Alignment of vorticity and scalar gradient with strain rate in simulated Navier-Stokes turbulence, Phys. Fluids 30:2343 (1987). Z. S. She, E. Jackson, and S. A. Orszag, Structure and dynamics of homogenous turbulente: Models and Simulation, Proc. R. Sec. Lond. A. 434:lOl (1991). B. J. Cantwell, Exact Solution of a restricted Euler equations for the velocity gradient tensor, Phys. Fluids A. 4:782 (1992). A. J. Majda, Vorticity, turbulente, and acoustics in fluid flow, Siam Rewiew 33:349 (1991). P. Vieilletosse, A Three-dimensional integral method for calculating incompressible turbulent skin friction, J. Physique 43:837 (1982). W. C. Reynolds, Measurements in fully developed turbulent channel flow, Ann Phys. Fluid Mech 3:183 (1976). K. Rajendran, Simulation of unsteady turbulent flows by quasisteady models, Appl Math Comput 48:l (1992). D. P. Telionis, Unsteady Viscous Flows, Springer-Verlag, New York, 1981. 0. R. Burggraff, Comparative study of turbulente models for boundary layers and Walls, ARL TR 74-0031 (1981).

110 11 12 13 14 15 16 17

18

19 20

K. RAJENDRAN T. Cebeci and H. B. Keller (Eds), Recent Research in Unsteady Boundary Layers. Vol. 11, pp. 1072, 1972. D. E. Abbott and F. Cebeci (Eds), Fluid Dynamits of Unsteady 30 and Separated Flows, pp. 202, 1971. T. Cebeci, Skin-friction characteristics of laminar power-law fluids on a slender circular cylinder. AIAA J. 8:2152 (1970). P. P. Telionis and D. Th. Tsahalis, Oscillatory laminar boundary layers and unsteady Separation, AIAA J 12:1469 (1975). P. Bradshaw, Calculation of boundary layer development using the turbulent energy equation, NPL AERO Rept., VI:288 (1969). H. A. Dwyer, E. D. Doss and A. L. Goldman, A Computer program for the calculation of laminar and turbulent boundary flows, NA CA CR:4366 (1970). R. N. Gupta and R. L. Trimpi, Influence of induced turbulente on the hydrodynamics of a wall boundary layer, Jpn Sec. Mech Engg. Sec. Chem Engg. 2:339 (1974). T. Cebei, Calculation of unsteady two-dimensional laminar and turbulent boundary layer with fluctuation in external velocity, Proc. R. Sec. Lond A. 355:225 (1977). W. C. Reynolds, Computation of turbulent flows. Ann Reu. Fluids Mech 8:183 (1976). R. E. Singleton and J. F. Nash, Method for calculating unsteady turbulent boundary layers in two and three dimensional flows. Proc. AIAA Comp. Fluid Dyn Conf. p. 84 (1973).