effect of a single roughness element on boundary layer transition over a wedge

Effect of a Single Roughness Element on Boundary Layer Transition over a Wedge

Takashi Watanabe Department of Mechanical Engineering, Iwate University, Morioka, Japan

Ryoji Kobayashi Department of Mechanical Engineering 11, Tohoku University, Sendai, Japan

I

BAn experimental study was made to examine the effect of a tripping wire on boundary layer transition over a wedge of 15 °, 30 °, or 60 ° with pressure gradient. Turbulence measurements were carried out in a range from the near-wake of the wire to the fully turbulent region. The results were as follows: (1) The transition Reynolds number Re t (= U~xt/u) at the most effective condition is given by the formula Re t = 1.06 Re k + (1.33 × 104), where Rek (= Uoox k/u) is the Reynolds number at the wire position, U~ the free-stream velocity, ~ the kinematic viscosity of the fluid, and x k and x t the location of the wire and the transition point measured from the apex of the wedge, respectively. (2) The minimum transition Reynolds number Retm is a function of the ratio of the height of the roughness element k to the displacement thickness 6" of the boundary layer at the location of the element. The magnitude of Retm decreases with increasing k/6* and increases with the increasing wedge angle at a fixed value of k/6*.

Keywords: boundary layer control, transition flow, wedge flow, pressure gradient, surface roughness

I N T R O D U C T I O N

A condition of the boundary layer transition from laminar to turbulent flow is remarkably influenced by free-stream turbulence, pressure gradient, and surface roughness. One way to promote the transition is to introduce direct disturbances into the boundary layer. This can be achieved by means of a tripping wire.

The boundary layer transition on a smooth solid surface occurs at some distance x o from the leading edge. When a two-dimensional roughness element is placed on the surface spanwise and perpendicular to the flow direction, the transition occurs at a location x t less than x 0. The location x t of transition is a function of the height k of the roughness element, its location x k, the free-stream velocity U~, and the viscosity ~ of the fluid. There are many experiments on the transition Reynolds number for a fiat plate with zero pressure gradient with the tripping wire attached to the fiat plate [!]-[6] and with a wire fixed at a small distance away from the plate [7]-[10]. The effect of the tripping wire on flow around a cylinder in the presence of a pressure gradient has also been investigated [11, 12]. However, the relation between the effect of the tripping wire on the boundary layer transition and the pressure gradient, and the transition process due to the wake of the tripping wire, have not been sufficiently clarified.

The purpose of this paper is to make clear the mechanism of the boundary layer transition along a wedge having a pressure gradient by means of a tripping wire. The main reason for this is that the promotion of the laminar to turbulent transition on the wedge surface greatly increases heat transfer and reduces the form drag, which are of signif- icant practical importance to engineering applications. As

reported in this paper, experiments were done with a two- dimensional roughness element by means of a wire of circular cross section attached to the wedge surface. Treatment is for the cases of wedge angle f~ -- 15 °, 30 °, and 60 °, and it is clear that the effects of the tripping wire on the boundary layer transition are changed by the pressure gradient.

E X P E R I M E N T A L M E T H O D A N D E Q U I P M E N T

The experiments were carried out using a wind tunnel of the return circuit type. The contraction nozzle of the wind tunnel has an area reduction rate of 4. The exit of the contraction nozzle is of square cross section 0.6 m × 0.6 m, and the length of the test section of open-jet type is 0.9 m. The turbulence intensity of the mainstream at the test section was kept less than 0.2% of the flow velocity throughout the experiments. Figure 1 shows a test model of the wedge, and Table 1 gives the designed values of wedge angle ri = 15 °, 30 °, and 60 ° and the actual measured values rio; the length of the slant surface of the wedge, ll; and the base length of the wedge, 12. The wedge length in the span direction is 170 mm, and a spanwise variation of the turbulent transition distance L from the wire element location to the transition point is shown in Fig. 2 for fl = 30 °, Re k = 1.30 × 105 , x k = 100 mm, and k = 1.0 ram. From Fig. 2 the distance L is changed at both sides of the wedge due to the outer edge of the main airstream but remains constant at the central width of 100 mm on the wedge surface. Although such a constant width was somewhat varied with experimental conditions, two-dimensional characteristics for the transition point were maintained in the central region of the wedge surface. Results

Address correspondence to Professor T. Watanabe, Department of Mechanical Engineering, Faculty of Engineering, Iwate University, Ueda 4-3-5, Morioka 020, Japan.

Experimental Thermal and Fluid Science 1991; 4:558-566 © 1991 by Elsevier Science Publishing Co., Inc., 655 Avenue of the Americas, New York, NY 10010 0894-1777/91/$3.50 558

/3o0

l J ~ 2 3 _ _

Figure 1. Schematic diagram and coordinate system.

Table 1. Values of Wedge Body

12 15 * 30* 60* 12o 15.1" 30.2* 60.1" I t 305 mm 305 mm 230 mm 12 160 mm 240 mm 230 mm n 0.0435 0.0910 0.2050

similar to the one in Fig. 2 were obtained under the present experimental conditions of the wedge angle fl, Reynolds number Reg, wire location xk, and wire diameter k. The wedge is supported on the downstream side by a bar of diameter 34 mm. To consider a surface without a roughness effect, the wedge surface was ground to a high degree of smoothness. In order to eliminate any clearance or gap between the wedge surface and the roughness element of circular cross section, an instant glue was used to attach the element to the surface. The range of the element location x k varied from 25 to 200 mm, and the wire diameter k was 0.5 mm and 1.0 mm. The velocity field in the boundary layer downstream of the tripping wire was measured by an I-type constant-temperature hot-wire anemometer. The uncertainty

of local velocity fluctuation intensities x / ~ / U ~ o was estimated at 1%.

E X P E R I M E N T A L R E S U L T S A N D D I S C U S S I O N

Veloci ty o f Outer Edge o f B o u n d a r y Laye r a n d Di sp l acemen t Th ickness

Figure 3 shows the profile of velocity U e at the outer edge of the boundary layer along the inclined surface of the wedge in the x direction. The vertical axis is in the dimensionless expression Ue/Uoo. The broken lines in Fig. 3 represent theoretical velocity profiles of potential flow along the wedge. The velocity deviation of the experiment from the potential flow might come mainly from the velocity difference between the outer edge of the boundary layer and the wedge surface.

Table 1 shows the corresponding values of n for wedge angles fl = 15", 30", and 60*. The velocity along the inclined surface is expressed as U e = ax n, where a is a constant and n is a function of the wedge angle ft. The constant a was determined under the condition that the value of U e / Uoo at x = 0.1 m becomes the same as the experimental value.

Effect of a Single Roughness Element 559

15 ~ i I I I J l

I 0 S £,

5 I I I I I I I I - 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 8 0

Z []m

Figure 2. Turbulent transition distance L in span direction z. 12= 30 ° ,Re k = 1.30 × 105 , k = 1.0mm, and x k = 100ram.

The displacement thickness 6" of the laminar boundary layer in the absence of the roughness element at the wedge surface is shown in Fig. 4. The results were obtained by calculation in the case of free-stream velocity Uo. = 10 and 20 m/s when the wedge angle was varied, f] = 15", 30", and 60".

Turbu lence Fie ld D o w n s t r e a m of the T r ipp ing Wi re

For the purpose of investigating the aspect of flow behind the roughness element, an experiment was carried out under the following conditions: included angle of wedge fl = 15 °, wire diameter k = 1.0 mm, element location x k = 200 mm, and free-stream velocity U~ = 13.8 m/s. The Reynolds number Re k turned out to be 1.81 × 105 (Re k = U=Xk/~,) , and the displacement thickness /~* of the laminar boundary layer at the element location was found to be equal to 0.68 mm. The results are given in Figs. 5 -9 .

Figure 5 shows the profile of velocity fluctuation intensity

V~u2 /U~ in the y direction and the ratio X / k as parameter, where X is the distance from the element location and k is the wire diameter. The position of the maximum velocity fluctuation is shown by the broken line Y , n / k . The uncertainty of the Y m / k curve was estimated at +_ 0.01. We know that when the ratio X / k increases beyond 3, the position of Y m / k rapidly decreases below 0.5, whereas it increases if the ratio increases beyond 14. As the velocity fluctuation increases, it spreads entirely within the boundary layer and

I . I I I i ]

30 °

~ 1.0

~ 0.9

. ~ / / z - - . & 6 0 o - - E x p e r i m e n t

0.8 / / . . . . . } Theory

/ l i I 0 .70 0.~)5 O. I OJ 5 0.2

X m

Figure 3. Ve loc i t y d is t r ibu t ion U e at the outer edge o f the boundary layer in the x d i rect ion. 12 = 15 °, 30 °, and 60 ° .

561) Takashi Watanabe and Ryoji Kobayashi

6<3

5 0 °

0.5- ~

~; '~ IJ,,, : i o m/s I

(,,,,= 2o m/sj 0 I . . . . I

0 O.i 0 .2 0 .3 .Z m

Figure 4. Displacement thickness 6" of the boundary layer in the x direction. (2 = 15", 30", and 60*; U= = 10 m/s , 20 m/s.

advances downstream. Results similar to those in Fig. 5 were obtained for different values of Re k.

Figures 6 - 9 show the oscil lograms of velocity fluctuation and the frequency spectra at selected positions in Fig. 5. First, the oscil logram of velocity fluctuation and frequency spectrum Su is shown in Fig. 6 at a position comparatively close to the wire, X = 3k and y/k = 0.5. From the figure the velocity fluctuations of the sine curve occur at a frequency of about 3.7 kHz. These velocity fluctuations might come from vortices shedding from the top of the wire. It is shown in Fig. 7 that as the distance X from the element increases, the amplitude o f velocity fluctuation also increases but the velocity fluctuation remains periodic. The value of y at which the velocity fluctuation becomes maximum reduces to 0 . 2 k as shown in Fig. 5. The frequency spectra at X = 12k for y/k = 0.3, 0.5, and 1.0 are shown in Fig. 8. The spectra include high-frequency components, which means that the location is near the fully turbulent state. Mild peaks

1.5

1.0

0.51 0

0

- - 1 I I

t6

I I

2 4 6

~/U~o %

Figure 5. Distribution of velocity fluctuation intensity x/~ /U= in the y direction. (2 = 15", Uoo = 13.8 m/s , Re k = 1.81 x l 0 5 , k = 1 .0mm, x k = 200mm.

i ' i ' i '

AAAAAAA A A nAAAA/I

| , , i I

0 I 2 3 4

Time ms

10 2 10 3 10 4

F r e q u e n c y J: Hz

Figure 6. Oscillogram of u fluctuation and frequency spectrum. (2 = 15 °, X = 3k, y/k = 0.5, k = 1.0 mm, x k = 200 ram, Re k = 1.81 × 105 .

remain in the range of 1.8 kHz. Figure 9 shows the velocity fluctuations and the corresponding frequency spectrum for y/k = 0.5 at X = 16k. Results similar to those in Fig. 9 were obtained at Re k = 1.81 x 105 when the position of y was changed, that is, y/k = 0.3, 0.7, and 1.0, and when the distance X from the element became more than 16k. When the distance X increases beyond 16k, the turbulent range spreads over the boundary layer, and accordingly a turbulent boundary layer forms there. Referring to the above, we determined the turbulent transition location T for different Reynolds numbers Re k at y/k = 0.5 by using the frequency spectra of velocity fluctuations. The uncertainty of determin- ing the transition location T at y/k = 0.5 was estimated at +1 m m o f L.

nAAAAA AAAnAAI JVVVVVV vvvvVv I ! I

0 2 4 6 8

Time ms

10 2 10 3 lO 4

Frequency .f Hz

Figure 7. Oscillogram of u fluctuation and frequency spectrum. fl = 15", X = 10k, y/k = 0.3, k = 1.0 mm, x k = 200 mm, Re k = 1.81 x l05.


4

10 2 10 3 10 4

F r e q u e n c y ~ Hz

Figure 8. Frequency spectrum, fl = 15", X = 12k, k = 1.0 mm, x k = 200 mm, Re k = 1.81 × 103 .

Figure 10 shows the maximum values of velocity fluctuation at each position of X / k for different values of Re k. The turbulent transition location T is indicated by arrows in Fig. 10. It is clear from the figure that the intensity of velocity fluctuation increases with increases in the distance downstream from the element and continues to increase beyond the point T even if the turbulent condition within the boundary layer is attained. After that the maximum value of velocity fluctuation gradually reduces. The point X / k of the maximum value of the curves decreases as Re k increases. As Re k increases, the point X / k of the maximum value approaches the element, with transition occurring instantly, and the magnitude of this maximum value becomes large. Results similar to the above experimental results for k = 1.0 mm and x k = 200 mm were obtained for different values of the element location x k and wire diameter k.

Transition Reynolds Number Re t

The relations between the distance L measured from the location of the roughness element to the turbulent transition point and the free-stream velocity /..7= are shown in Figs. 11-13. Figure 11 shows the results for 12 = 15", two ele-

T T 1

2 4 6 8 Time ms

10 z 10 3 10 4

F r e q u e n c y :f Hz

Figure 9. Oscillogram of u fluctuation and frequency spectrum. 12 = 15", X = 16k, y / k = 0.5, k = 1.0 mm, x k = 200 mm, Re k = 1.81 x 105 .

I 0 - - I i r - t

Re,~ = 2 . B I x l 0 s

O 1 / ~ ~ / I ' O I x I 0 5

A 6

4 i . i T x l O S

0 / ~ ~ I I 0 tO 20 3 0 4 0 5 0

x / k

Figure 10. Variation of the maximum velocity fluctuation inten-

sity (V/~/U=)max in the X direction, fl = 15", k = 1.0 mm, and x k = 200 mm.

ment locations x k = 50 and 200 mm, and four wire diame- ters k as parameter, while Fig. 12 is for 9 = 15", k = 0.5 and 1.0 mm, and x k as parameter. Figure 13 is for k = 0.5 and 1.0 mm, x k = 100 mm, and 9 as parameter. It can be seen from Figs. 11 and 12 that as U~. increases, the value of L decreases because the transition point T approaches the element location from downstream. When the free-stream velocity U= is large enough, the value of L tends to become constant. At a definite /.Jr=, the value of L becomes higher for increased x k. We also know that the turbulent transition begins at smaller free-stream velocity U= as the wire diameter increases. The arrow in Fig. 12 indicates the measuring condition (Re k = 1.81 x 105) of Fig. 5.

Figure 13 shows how the value of the transition distance L changes in accordance with the wedge angle ft. It can be seen that as fl decreases, the turbulent transition begins at smaller free-stream velocity. The distance L of turbulent transition is largely influenced by the wedge angle and the location and diameter of the roughness element. Here, the errors in the experimental results for L were within + 1 mm.

Figure 14 was obtained from Figs. 11 and 12 for fl = 15". It shows the relationship between Re k at the location of the roughness element and Re t with the ratio x k / k as parameter. The broken line in the figure shows the experimental

.0 m k =0.5 mm

~ .~, O.Tmm~ ' ~ ___ z~=200 mm \ .....

1.4 mm I " I I

tO 15 20 2 5 30 U= m/s

Figure 11. Turbulent transition distance L with varying diameter k. fl = 150; x k = 50 and 200 mm; k = 0.5, 0.7, 1.0, and 1.4 mm.

562 Takashi Watanabe and Ryoji Kobayashi

; t ~ 2 r % \ ~, . . . . . / ~ : , . o , m ; : ' ,oo/~% I% .

\ X ioo '.,t ~. o~/b ~. ~ ~ ..15o

O I J I I 5 I 0 15 2 0 2 5 3 0

Uoo m/s

Figure 12. Turbulent transition distance L with varying location x k of roughness position. ~ = 15"; k = 0.5 and 1.0 mm; x k = 25, 50, 100, 150, and 200 mm.

results about the boundary layer along a flat plate obtained by Kraemer [4].

Figure 15 also shows the relationship between Re t and Re, at the element location for wedge angle 9 = 30* and 60 °. It is evident from Figs. 14 and 15 that as Re, increases, the effect of the roughness element reaches a definite relation, and the relation between Re t and Re, can be expressed by a straight line M for all values of f/ and x , / k . The expression for the straight line M becomes

R e t = 1.06 R e , + (1.33 × 104) (1)

The other straight line N means Ret = Re , and L = 0, that is, x~ = x , . The difference between the two straight lines M and N corresponds to the limiting value of L for large values of U= in Figs. 1 ! - 13 and is given by the expression

U ~ L

- R e t - R e , = 0.06 R e k + (1.33 × 104) /.,

= (1 .5-2 .5) × 104 (2)

in the present experimental range of Re, = (0.09-2.0) × 105. Kraemer's expression [4] for the boundary layer along a fiat plate was Re t - Re k = 2.0 × 104 at Re k = 1.0 × 105. Therefore, the present result for a wedge with pressure gradient includes the one for a flat plate without pressure gradient. In order to explain the relation between Re t and Re, in Figs. 14 and 15, a curve is denoted by symbols A B C for fl = 15 °, X k / k = 200, and k = 1.0 mm in Fig. 14 and for 9 = 30 ° , X k / k = 200, and k = 0.5 mm in Fig. 15. In the range of Re, between A and B, Re t lies on the straight line M. As Re, decreases further, Re t separates from the

I00

-4 ~ 0

0 5

'1 ' I [ ~ " " I I

k - , o . - ', I , , o . . . . ' , i .,o °

I I I I I 0 I 5 2 0 2 5 3 0

U® m/s

Figure 13. Turbulent transition distance L with varying wedge angle fL x k = 100mm; k = 0.5 and 1.0 ram; ~ = 15", 30", and 60*.

2.0 . . . . . . . . , . . . . . . . • . . . . . . , ~ ~ , / , : 2 o o ~ t / ~ :

% " A ! t J/ 4

, o o .

2 5 ~ ° 7 5 •

• r t o , o , , / ~ / N . . . . F l a t p l a t e [ 4 ]

O v~__ I 1 I 0 0 . 5 1.0 1.5 2 . 0

Rek Y, I 0 ~

Figure 14. Relation between transition Reynolds number Re t and wire position Reynolds number Re k. f/ = 15 °, k = 0.5 and 1.0 mm.

straight line M upwards along the curve BC. This means that the transition point T starts to move downstream. A further decrease in Re k results in an abrupt increase in Re t for small reduction of Re k near the point C.

Figure 16 and 17 show transition processes of velocity fluctuations in the X direction for two cases of pronounced wire effect on the straight line M and diminished wire effect on the curve B C in Fig. 15, respectively. For the pronounced wire effect (Re k = 1.75 × 105), Fig. 16a shows the velocity fluctuations of the sine wave released from the roughness element. As distance X from the roughness be-

2 . 5 r . . . . . . i T . . . .

~o .i/7, / k : 2 0 0 x

1.5 I o 0 i ,

1 M i

0 . 5 ,Q, 3 0 ° 6 0 °

k = 0 . 5 ,m I " / "

Re, =Re, k=l.Omm i o ] o 0 / Re, : Re, I ,

o 0.5 i .o i. 5 Rek x I0 ~

Figure 15. Relation between transition Reynolds number Re t and wire position Reynolds number Re k. ft = 30* and 60*, k = 0.5 and 1.0 mm.

I i I

IA t l l tAAAAAAAAAAz%A,~AAAAP, I I A t ~ A h A A A A A VVVUVVVVVVVVVV v V V V V V V V V V V V V V V V V V

( a ) X l k = 4 i i i

t,/A A,~,\ AA~ A ,, AAA,.,.AAA t l, vvvvvv,vv - vvvv vv -- I i

( b ) X / k = 8 i i

V /Vu'W V v. F .7, (c ) X/k=12

( d i

) X / k = I 6 i i

'VVVP |

I l l s

ll, /1

i

Time 0 1 2 3 4

( e ) X / k = 2 0

Figure 16. Oscillograms of u fluctuation showing development of turbulence, fl = 30*, Re k = 1.75 x lO 5, k = 0.5 mm, and x k = 100 mm.

comes large, the amplitude increases but the velocity fluctuation remains periodic (Fig. 16b). High-frequency components are gradually included in the velocity fluctuation (Figs. 16c, 16d), and the boundary layer becomes fully turbulent (Fig. 16e).

Figure 17 gives the velocity fluctuation of the transition area in the case of the diminished wire effect on the curve B C for Re k = 1.30 x 105. The sinusoidal fluctuation in Fig. 17a indicates periodic shedding of vortices from the wire element. The amplitude of the velocity fluctuation decreases with increases in X (Fig. 17b). Fresh fluctuation waves appear again as X increases further (Fig. 17c); the growth of these fluctuations results in transition to turbulence (Figs. 17d, 17e). Figures 16 and 17 show the case for wedge angle

Effect of a Single Roughness Element

, I I

IAAAAAA.AA.AAA/\AAAAAAI IV VVVVVVVVVVVVVVVVV,,, 0 1 2 3 4

( a ) X / k = 6

. ~+ ,^ . , , ~ ,~ .... i < , ~ ° ^,,%o,'. . . t

( b ) X / k = 1 6

$63

r

( c ) X / k = 2 4 I I

( d ) X / k = 3 6 i

Time

II AI ,,

0 15 I S

( e ) X / k = 5 2

Figure 17. Oscillograms of u fluctuation showing development of turbulence, fl = 30", Re k = 1.30 × 105 , k = 0.5 mm, x k = 100 mm.

= 30*, element location x k = 10 mm, and wire diameter k = 0.5 mm. Further experiments showed that the difference between the transition processes of Figs. 16 and 17 is similar for other values of fl, k, and x k.

Figure 18 shows how the magnitude of the velocity fluctu-

ation x / ~ / U= at y / k = 0.5 changes downstream for wedge angles of fl = 15", 30", and 60", wire diameter k = 0.5 mm, and element location x k = 100 mm. The solid lines show the case of R% = 1.75 x 105 on the straight line M in Figs. 14 and 15 where the effects of the wire are well


o~ 8 - - I ~ 1 1 - ~ . . . . . . . . • ~ , = 1 5 ° • o

4

Z,/¢ • ij /

0 2 0 4 0 6 0 8 0 I 0 0 × / k

Figure 18. Variation of the velocity fluctuation intensity v/-~-/U= in the X direction. Re k = 1.30 × 105 and 1.75 x 105 , y / k = 0.5, k = 0.5 mm, x k = 100 mm.

pronounced, and the broken line shows the case of R% = 1.30 × 105 on the curve BC where the effects of the wire are diminished. For Re k = 1.75 × 105, the free-stream velocity was U= = 25.8 m/ s and the boundary layer displacement thickness 6* at the element location for ~ = 15", 30 °, and 60* was equal to 0.37, 0.33, and 0.28 mm, respectively, while for Re k = 1.30 × 105 , Uoo was 20.2 m/s , and 6* at f~= 15 °, 30 °, and 60 ° was 0.41, 0.37, and 0.31 mm, respectively. As seen in Fig. 18, the magnitude of velocity

fluctuation x~u2/U~o at R% = 1.75 x 105 increases directly with X and reaches a turbulent state at X = 20k. The transition Reynolds number for X = 20k is Re t = 1.89 x

105. On the other hand, the magnitude x~u2/U= at Re k = 1.30 × 105 for f~ = 30*, for example, increases slightly to X = 8k and then decreases from X = 8k to X = 16k.

From X = 16k to X = 24k , the magnitude of ,~u2/U= remains nearly constant. When X becomes larger than 24 k, fluctuations of high-frequency components and the magnitude

of ~ / U o o increase gradually. At X = 52 k, the transition region transfers to the turbulent state. The transition Reynolds number for X = 52k is Re t = 1.64 × 105. The transition process for wedge angle t2 = 60 ° is similar to that for f/ = 30 °. At X = 80k , the transition region transfers to the turbulent state, which corresponds to Re t = 1.82 × 105. The arrows in Fig. 18 indicate the turbulent transition point T. The distance L between the wire location and the transition point T for Re k = 1.75 × 105 is almost constant even if the wedge angle changes. This condition corresponds to a high value of Uoo in Figs. 11-13. As the wedge angle t2 increases at Re k = 1.30 × 105, Re t increases, which means that the transition point T moves away from the wire location.

Figure 19 represents values of Rekl, at which the tripping wire has no effect on the boundary layer transition, in relation t o Xk/k for wedge angles ~ = 15 °, 30 °, and 60 °. The values of Rex1 were roughly obtained as the point C in Figs. 14 and 15. We know that Rek~ depends on the dimensionless value x k / k even if k and x k are varied and that as x k / k becomes large, Rekl increases along the straight line.

As stated in Figs. 14 and 15, Re t could be expressed in the linear relation of M to Re k in the case where the wire effect is well pronounced even if f] or k is changed, while Re t could be expressed in the curve BC for the diminished wire effect and has a min imum value Retm of Re t.

1.5

3o

× i J k : o . 5 m m " I " i / t

Y ,/4

0 . 5

0 I o IOO 2 0 0

z~ / k

Figure 19. Relation between x k / k and Reynolds number Rekj at which the tripping wire has no effect on transition, k = 0.5 and 1.0 ram; fl = 15", 30". and 60".

Figure 20 expresses the relation between Retm and x k / k obtained from Figs. 14 and 15. The value of Retm is clearly dependent on Xk/k , and as Xk/k becomes large, Re, , , increases along the straight line. It is seen that Retm increases as fi increases and the wire diameter k decreases. Thus the change of R e , , is controlled by fi and k.

2 . 0 I J I - - r %

• , , o Q. = 60"

• ,o D , : 3 0 °

,.5 . . . . . .£/. = , 5" / . / " / / ~ . ' " o ~ . / / o

m 0 . 5 -

0 [ I t t o 50 ,oo ,50 zoo

z~ /k

Figure 20. Relation between minimum transition Reynolds number Retm and xe/k . 12 = 15", 30", and 60*; k = 0.5 and 1.0 mm.

~o_x 2,51 ----r . . . . . [

¢7 2"0 I

1.5

1.0

0.5

i l I

+

Re,,~ ~ • k =o.5 ®m o k=l. Omm

zk / k = 20o

150

Ioo

a Plat plate[6] 0 i I I

0 I 2 5 4 k / 6 *

Figure 21. Transition Reynolds number Re t and ratio of wire diameter to boundary layer displacement thickness k/b*. 12 = 15", k = 0.5 and 1.0 mm.

Figure 21 shows the relation between Re t and k / 5 * , which is the ratio of the wire diameter k to the boundary layer displacement thickness tS*, with x k / k as parameter for fl = 15". The above relation was obtained by using Fig. 4 for the displacement thickness tS* of the laminar boundary layer on the wedge without a roughness element. In Fig. 21, a solid line and a broken line give the relation between k/(5* and the corresponding minimum value Retm for k = 1.0 and 0.5 mm, respectively. As k /~* increases, Retm decreases remarkably. The broken line represents the experimental results of the boundary layer over a flat plate [6]. Compared with the results of the present experiments for overlapping k /5* there is good agreement qualitatively and a fairly close relation quantitatively between the two curves. Figure 22 shows the relation between Retm and the ratio of wire diameter k to boundary layer displacement thickness +5* for fl as parameter. It is evident from the figure that Retm increases as fl becomes large.

P R A C T I C A L U S E F U L N E S S / S I G N I F I C A N C E

Fundamental research was conducted to clarify the effect of two-dimensional roughness on boundary layer transition in the presence of a pressure gradient in detail. Promoting the turbulent transition is closely related to reducing form drag and increasing heat transfer. Therefore, the present experimental results are valuable to a basic understanding of the transition process. For example, in Figs. 16-18, the velocity fluctuations in the transition process and the change of magnitude of the velocity fluctuations, in the cases of pronounced wire effect (Re k = 1.75 x 105) and diminished wire effect (Re k = 1.30 × 105), respectively, were shown. The relations between the transition Reynolds number Re t and the Reynolds number Re k at element location x k were given in Fig. 15 and in Eq. (1). The Reynolds number Rekt, at which the tripping wire has no effect on boundary layer transition,


2.0

%

i 1.5

1.0

0.5

I .... k= o.5 mm

k : l.omm

%'%.%

\ ' . . _ _

15 °

0 I 1 2 3 4

k / a * Figure 22. Minimum transition Reynolds number Ret, " versus ratio of wire diameter to displacement thickness k/~i*, fl = 15", 30", and 60*; k = 0.5 and 1.0 mm.

and the minimum transition Reynolds number Retm in relation to x k / k were shown in Figs. 19 and 20, respectively. In Fig. 22, the relationship between Retm and the ratio of wire diameter k to boundary layer displacement thickness 6* was shown for various values of the pressure gradient. These relations are useful for checking the turbulent transition on more complex models.

C O N C L U S I O N S

Experimental investigations were carried out for wedges with three pressure gradients in uniform flow to study the effect of a tripping wire on the turbulent transition of a boundary layer. The following results were obtained.

1. The velocity fluctuation field downstream of an isolated roughness element produced by a wire of diameter k was made clear. It showed that the maximum value of the velocity fluctuation appears rather near the wedge surface, that is, at a location less than the height of the wire diameter.

2. The magnitude of velocity fluctuations due to the roughness element increases continuously in the downstream direction even if the turbulent transition point T is ex- ceeded and then decreases gradually after the maximum value.

3. In the case where the wire effect is well pronounced, the relation between the transition Reynolds number Re t and the Reynolds number Reg at element location x k remains on a straight line even if the wedge angle fl and the wire diameter k are changed. The relation can be expressed as Re t = 1.06 Re k -F (1.33 × 104).


4. The relation between the Reynolds number Rekx, where the tripping wire is of no effect, and the wire location x k / k is shown in Fig. 19. This relation could be given by a straight line where the value of Rekl increases with increasing X k / k . It is evident that Reki is increased as the wedge angle fi increases and the wire diameter k decreases.

5. The min imum value of the transition Reynolds number, Retm, decreases with increasing values of k / S * . At a fixed value of k / S * , Retm increases with increasing values of the wedge angle ft.

NOMENCLATURE

f frequency of velocity fluctuations, Hz k height of roughness element (wire diameter), mm L distance from element location to transition point

( X t - - Xk) , m m

m pressure gradient parameter, dimensionless Re k wire position Reynolds number ( = U~x~/p ) ,

dimensionless Re t transition Reynolds number ( = U ~ x , / ~ ) ,

dimensionless u velocity component in the x direction, m / s

U~ velocity of free stream, m / s U e velocity at outer edge of boundary layer, m / s X distance from roughness element, mm x distance from leading edge of wedge, m

x k distance from leading edge to roughness element, mm x t distance from leading edge to transition point, mm y distance from surface of wedge, mm z distance from centerline to side of wedge surface in

span direction, mm

Greek Symbols kinematic viscosity, m2/s

c5" boundary layer displacement thickness, mm fi total angle of wedge, degrees

REFERENCES

1. Tani, I., Hama, R., and Mituisi, S., On the Permissible Roughness in the Laminar Boundary Layer, Rept. Aeronautical Research Inst., Tokyo Imperial Univ., No. 199, 1940.

2. Tani, I., and Hama, F. R., Some Experiments on the Effect of a Single Roughness Element on Boundary-Layer Transition, J. Aero- nautical Sci., 20, 289-290, 1953.

3. Dryden, H. L., Review of Published Data on the Effect of Rough- ness on Transition from Laminar to Turbulent Flow, J. Aeronauti- cal Sci., 20, 477-482, 1953.

4. Kraemer, K., /.)ber die Wirkung von Stolperdr~ihten auf den Gren- zschichtumschlag, Z. Flugwiss., 9, 20-27, 1961.

5. Klebanoff, P. S., and Tidstrom, K. D., Mechanism by Which a Two-Dimensional Roughness Element Induces Boundary Layer Transition, Phys. Fluids, 15, 1173-188, 1972.

6. Schlichting, H., Boundary Layer Theory, 7th ed., McGraw-Hill, New York, 1978.

7. Okamoto, T., and Takeuchi, M., Effects of Side Walls of Wind- Tunnel on Flow Around Two-Dimensional Circular Cylinder and Its Wake, Trans. Jpn. Soc. Mech. Eng., 41, 181-188, 1975. (In Japanese.)

8. Bearman, P. W., and Zdravkovich, M. M., Flow Around a Circular Cylinder near a Plane Boundary, J. Fluid Mech., 89, 33-47, 1978.

9. Kanemoto, T., Toyokura, T., and Kurokawa, J., Inference Between Boundary Layer and Turbulent Wake, Trans. Jpn. Soc. Mech. Eng., 46, 1237-1244, 1980. (In Japanese.)

10. Muraoka, K., Effect of Cylinder with a Gap to a Flat Plate on Boundary-Layer Transition, Trans. Jpn. Soc. Mech. Eng., 50B, 59-67, 1984, (In Japanese.)

11. Fujita, H., Takahama, H., and Kawai, T., Effect of Tripping Wires on Heat Transfer from a Circular Cylinder in Cross Flow, Trans. Jpn. Soc. Mech. Eng., 50B, 1275-1284, 1984. (In Japanese.)

12. Igarashi, T., Effect of Tripping-Wires on the Flow Around a Circu- lar Cylinder Normal to an Airstream, Trans. Jpn. Soc. Mech. Eng., 52B, 358-366, 1986. (In Japanese.)

Received August 30, 1990; revised January 12, 1991

effect of a single roughness element on boundary layer transition over a wedge

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