WO2013187830A1

WO2013187830A1 - Miniature vortex generator

Info

Publication number: WO2013187830A1
Application number: PCT/SE2013/050659
Authority: WO
Inventors: Jens FRANSSON
Original assignee: Fransson Jens
Priority date: 2012-06-11
Filing date: 2013-06-10
Publication date: 2013-12-19

Abstract

A miniature vortex generator for delaying transition to turbulence in a flow of a fluid over a surface in vicinity of a leading edge, wherein the fluid has a kinematic viscosity v and the flow has a free-stream velocity U ∞, wherein the free-stream velocity varies in a streamwise direction of the flow approximately 5 as a power function of the distance from the leading edge with a leading power m. The miniature vortex generator comprises an array of substantially plate shaped elements protruding from the surface. Each of the elements is arranged at a distance from the leading edge in the streamwise direction, and each of the elements is arranged at an angle in the range from 9 to 18 or from 10 6 to 15 degrees relative to the streamwise direction. The height of each of the elements is non-decreasing with respect to the streamwise direction. The delay of transition turbulence has been observed only in fairly narrow height intervals depending on U ∞, v and m.

Description

MINIATURE VORTEX GENERATOR Technical Field of the Invention

The invention relates to the field of fluid mechanics. In particular, the invention relates to a passive device for controlling boundary layer transition and the use of a device for controlling boundary layer transition.

Technical Background

The flow around a body, like the air around an aircraft or the water around a tank ship, produces a major drag force which leads to an increased energy consumption.

In general, the largest drag component is linked to the pressure distribution around the body and is called form drag. Its value is strongly correlated with the dimension of the wake that is generated due to the separation of the boundary layers formed on the body surface.

For streamlined bodies moving in fluids, a viscous drag is generated in the thin boundary layer surrounding the body skin, where the velocity of the fluid decreases to zero at the surface. In the upstream part of a streamlined body, the boundary layer flow is laminar: its velocity profiles are steady and stable to the small perturbations introduced by imperfections of the skin, turbulence in the incoming flow, and other noise sources. It is, however, well known that

U ^■ x sufficiently far downstream (where the Reynolds number Re =—— based v

on the incoming velocity U_∞, the downstream distance from the leading edge x and the kinematic viscosity v of the fluid, exceeds a critical value) the flow becomes unstable and small perturbations can be amplified, leading to transition to turbulence when they exceed a critical finite amplitude. Owing to the increased shear stresses, the mean skin friction of a turbulent boundary layer is typically an order of magnitude larger than that of a laminar flow. For a "clean" base flow, i.e. a flow with low background disturbance levels typically encountered in free flight, and a hydraulically smooth surface, the classical transition scenario takes place with exponentially growing disturbance modes. The least stable mode, denoted Tollmien-Schlichting (TS) wave, starts to grow at a critical Reynolds number.

A known technique to prevent flow separation, and to thereby reduce form drag, is to introduce streamwise vortices through vane-type passive vortex generators inside the boundary layer. Such devices are commonly used on aircraft wings. This type of vortex generator normally consists of a row of blades or airfoils mounted perpendicular to the surface and with an angle against the oncoming flow. The height of these blades is often slightly greater than the boundary layer thickness.

It has been the common belief for decades that any type of surface roughness is negative with respect to turbulence in the sense that transition to turbulence is advanced, i.e. that the flow transitions to turbulence at a lower free stream velocity, thereby increasing the viscous drag. Fransson et al. (Delaying Transition to Turbulence by a Passive Mechanism, Physical Review Letters 96, 064501 , 2006) have shown however that transition to turbulence in the boundary layer may actually be delayed, i.e. that the flow transitions to turbulence at a higher free stream velocity, under certain circumstances by using circular roughness elements arranged on the surface, thereby reducing the viscous drag. Furthermore, Fransson (Afrodite, Proposal no. 258339 to the Seventh Framework Programme, 2010) has reported on indications that miniature vortex generators, i.e. vertical blades protruding from a surface, similar to traditional vortex generators but much smaller in size, may be even more effective than the circular roughness elements for delaying transition to turbulence. At present, it is neither known under which conditions this delay of transition to turbulence is achieved, nor how the miniature vortex generators should be designed in order to achieve the desired effect.

Summary of the Invention

It is an object of the present invention to provide design criteria for a miniature vortex generator for delaying transition to turbulence in a boundary layer on a surface, given a free stream velocity and a value of the viscosity of the fluid. According to a first aspect of the invention, there is provided a miniature vortex generator for delaying transition to turbulence in a flow of a fluid over a surface in vicinity of a leading edge, wherein the fluid has a kinematic viscosity v and the flow has a free-stream velocity U_∞ , wherein the free- stream velocity varies in a streamwise direction of the flow approximately as a power function of the distance from the leading edge with a leading power m, i.e. u_∞(x) = x^m + o(x^m) for small x. The miniature vortex generator comprises an array of substantially plate shaped elements protruding (in the tangential direction of the plates) from the surface. Each of the elements is arranged at a distance from the leading edge in the streamwise direction, and each of the elements is arranged at an angle in the range from 9 to 18 degrees relative to the streamwise direction. Alternatively, each of the elements is arranged at an angle in the range from 6 to 15 degrees relative to the streamwise direction. The height of each of the elements is non-decreasing with respect to the streamwise direction, and the maximum height h of each of the element fulfils:

A second aspect of the invention proposes the use of an array of substantially plate shaped elements protruding from a surface in vicinity of a leading edge of the surface for delaying transition to turbulence of a flow of fluid over the surface, wherein the fluid has a kinematic viscosity v and the flow has a free- stream velocity U_∞ , wherein the free-stream velocity varies in a streamwise direction of the flow as a power function of the distance from the leading edge with a leading power m, and wherein each of the elements is arranged at a distance from the leading edge in a streamwise direction of the flow, and wherein each of the elements is arranged at an angle in the range from 9 to 18 or from 6 to 15 degrees relative to the streamwise direction, and wherein the height of each of the elements is non-decreasing with respect to the streamwise direction, and wherein the maximum height h of each of the elements fulfils:

With a miniature vortex generator (MVG) according to the invention having elements with respective maximum heights within the above limits, transition to turbulence is delayed. The physical mechanism behind the stabilization appears to be related to the presence of a streaky base flow, i.e. extra turbulence terms are added to the base flow by the MVG, which stabilizes the boundary layer. The amplitude of the streaky base flow increases with the height of the elements of the MVG. If the height of the elements is too large, the streaky base flow becomes sensitive to instabilities which may advance transition to turbulence instead of delaying it. If the height of the elements is too small on the other hand, the effect of delaying the transition to turbulence is weakened and may vanish altogether. By having maximum heights of the elements fulfilling the above limits, the desired effect of delaying transition to turbulence may be achieved. The MVG is passive in the sense that it realizes a zero net energy exchange with the flow. No driving power is required.

In an embodiment of the invention according to the first or second aspects, each of the elements is arranged at an angle in the range from 6 to 9 degrees relative to the streamwise direction, or in the range from 9 to 15 degrees relative to the streamwise direction, or in the range from 15 to 18 degrees relative to the streamwise direction.

In another embodiment of the invention according to the first or second aspects, the maximum height h of each of the elements fulfils: m hU hU

Lower limit of — - Upper limit of — - υ υ

-0.09 362 470

-0.08 374 500

-0.05 423 546

0 478 608

0.09 558 710

0.2 634 807

0.5 797 1007

Within these limits, a particularly effective delay of the transition to turbulence may be achieved.

In yet another embodiment of the invention according to the first or second aspects, each of the elements is arranged a distance X_MVG downstream of the leading edge fulfilling:

By arranging the elements at a distance X_MVG from the leading edge fulfilling the conditions above, high amplitudes of the streaky base flow is achieved, thereby effectively delaying the transition to turbulence.

In yet another embodiment of the invention according to the first or second aspects, each of the elements is arranged a distance X_MVG from the leading edge fulfilling: m

Lower limit of ^XmvgU∞ Upper limit of ^XmvgU∞ υ υ

-0.09 4580 5150

-0.08 7500 9280

-0.05 18000 22300

0 46900 58100

0.09 152000 188000 0.2 409000 506000

0.5 2210000 2740000

It is understood that the present invention is not limited to the specific values of m and the corresponding values of the maximum height of the elements and the distances from the leading edge fulfilling the lower and upper limits tabulated above. On the contrary, the tabulated values should also be interpreted as a basis for interpolation for any intermediate values of m other than given above. It is also understood that the leading edge refers to the forward edge of the surface, i.e. the edge which is first encountered by a particle in the flow, e.g. the front edge of an airplane wing. It is furthermore understood that the streamwise direction refers to the direction of the flow, and that the spanwise direction refers to a direction perpendicular to the local streamwise direction. It is furthermore understood that plate shaped may refer to a flat body of constant thickness, but may also refer to a body having varying thickness but extending primarily in two dimensions. The thickness of the plate shaped bodies may for example be increasing in the streamwise direction or in the lengthwise direction of the bodies. Alternatively, the thickness of the plate shaped body may be greater close to the surface and decrease in the normal direction with the distance from the surface. It is furthermore understood that the feature of the free-stream velocity varying in a streamwise direction of the flow approximately as a power function of the distance from the leading edge with a leading power m refers to the fact that the velocity variation of the flow over the surface may approximately be described as a power function of the distance from the leading edge, where the leading power of the power function is m. The leading power m varies depending on the properties of the surface, e.g. roughness. A negative leading power indicates a decreasing free-stream velocity over the surface, and a positive leading power indicates an increasing free-stream velocity. It is also understood that the substantially plate shaped elements do not necessarily protrude from the surface at right angles with respect to the surface. In yet another embodiment of the invention according to the first or second aspects, the spanwise separations of the elements lie in the range from 5 to 17 times the height of the elements. In yet another embodiment, the spanwise separations of the elements lie in the range from 8 to 12 times the height of the elements. It is understood that the concept of spanwise separations refers to distances between the elements in the spanwise direction. The spanwise separations may be equal between all elements or may vary. By having spanwise separations of the elements within the above intervals, the transition to delay may be effectively delayed over the spanwise direction of the surface.

In yet another embodiment of the invention according to the first or second aspects, the elements are arranged in pairs and symmetrically with respect to the streamwise direction. In other words, a first element of each pair is angled in a first direction relative to the streamwise direction, and a second element of each pair is angled in a second direction, being opposite to the first direction, relative to the streamwise direction. Hereby, a symmetric boundary layer flow may be achieved and the stabilizing effect may be increased compared to when single elements with the same spanwise separation is used.

In yet another embodiment of the invention according to the first or second aspects, the spanwise intra-pair separations d of the elements of the pairs lie in the range from 0.75 to 1 .25 times the length of the elements. In yet another embodiment, the spanwise intra-pair separations d of the elements of the pairs equals the length of the elements. It is understood that intra-pair separation refers to the distance between the elements of each pair. The intra-pair separations may be equal for all pairs or may be different for different pairs.

In yet another embodiment of the invention according to the first or second aspects, the spanwise inter-pair separations D of the pairs lie in the range from 3.5 to 4.5 times the spanwise intra-pair separations d. In yet another embodiment of the invention according to the first or second aspects, the spanwise inter-pair separations D of the pairs lie in the range from 3.0 to 4.5 times the spanwise intra-pair separations d. In yet another embodiment, the spanwise inter-pair separations D equals 4 times the spanwise intra-pair separations d. It is understood that inter-pair separations refers to the distances between the pairs. The inter-pair separations may be equal between all pairs or may vary.

In yet another embodiment of the invention according to the first or second aspects, the thickness of each of the substantially plate shaped elements is less than the height of the element. In yet another embodiment, the thickness of each of the substantially plate shaped elements is less than half the height of the element.

In yet another embodiment of the invention according to the first or second aspects, the heights of the elements are increasing with respect to the streamwise direction. In yet another embodiment of the invention, the heights of the elements are linearly increasing with respect to the streamwise direction. In yet another embodiment, the elements have constant heights. In yet another embodiment, the heights of the elements are increasing

approximately as a logarithmic function with respect to the streamwise direction that is, it increases faster than linearly for small values.

In yet another embodiment of the invention according to the first aspect, the miniature vortex generator comprises a second array of substantially plate shaped elements, wherein the elements of the second array are arranged downstream the elements of the array. In yet another embodiment of the invention according to the second aspect, a second array of substantially plate shaped elements is used for further delaying the transition to turbulence, wherein the elements of the second array are arranged downstream the elements of the array. In yet another embodiment of the invention according to the first or second aspects, the elements of the second array are arranged in pairs, each of which is aligned in the streamwise direction with a

corresponding pair of the first array. In yet another embodiment of the invention according to the first or second aspects, the maximum heights of each of the elements of the second array are lower than the maximum heights of the elements of the array. Using a second array is advantageous in order to reinforce the stabilizing effect on the boundary layer, i.e. to further delay transition to turbulence. It is presently believed that the second array serves to regenerate the streaky flow downstream of the first array.

In yet another embodiment of the invention according to the first or second aspects, the substantially plate shaped elements are metallic blades which may be welded to the surface. Alternatively, the metallic blades may be attached to the surface by mounting means, such as for example mounting brackets, screw joints or rivets. In yet another embodiment, the substantially plate shaped elements are formed integrally with the surface in the form of a cast body. The cast body may be cast of a metallic material, such as aluminum or steel, or from plastic or a composite material. In yet another embodiment, the substantially plate shaped elements are formed integrally with a thin film adapted to be attached to the surface. The thin film may for example be plastic or metallic. It is noted that the invention relates to all combinations of features, even if these are recited in mutually different claims.

Brief Description of the Drawings

These and other aspects of the present invention will now be described in more detail, with reference to the appended drawing showing currently preferred embodiments of the invention, wherein:

fig. 1 shows a miniature vortex generator according to an embodiment of the present invention,

fig. 2 shows a miniature vortex generator according to another embodiment of the present invention,

fig. 3 is a side view of an element of a miniature vortex generator according to yet another embodiment of the present invention,

fig. 4 shows a diagram illustrating the upper and lower limits of the maximum height of the elements of a miniature vortex generator according to an embodiment of the invention,

fig. 5 shows a diagram illustrating the upper and lower limits of the distance from the leading edge to the elements of a miniature vortex generator according to an embodiment of the invention,

fig. 6 shows measurement data comparing boundary layer energy distribution with and without a MVG,

fig. 7a-b shows measurement data indicating the effect on the disturbance growth for different element heights at constant free-stream velocities, and

fig. 8a-b shows measurement data indicating the effect on the streak amplitude evolution in the streamwise direction.

Detailed Description

In the following description, embodiments of the present invention are described.

Fig. 1 shows a miniature vortex generator according to an embodiment of the present invention. The left part of figure 1 is a view from above showing a plurality of plate shaped elements which are protruding at right angles from a surface, i.e. in the y-direction. The plate shaped elements may also be described as blades. The elements are arranged at constant distances D apart in the spanwise direction, i.e. the z-direction. A main flow of fluid flows in the streamwise direction, i.e. the x-direction. All of the elements are arranged at an angle β relative to the streamwise direction. The elements are arranged along a straight line in the streamwise direction corresponding to a distance X_MVG from the leading edge of the surface (not shown). The right part of figure 1 is a side view of one of the elements. The element is triangular, i.e. a height which is linearly increasing from zero at the front edge pi to a maximum height h at the rear edge ph. The front edge pi is arranged upstream the rear edge ph such that the height of the elements is increasing in the streamwise direction. The element has a base length L. Fig. 2 shows a miniature vortex generator according to another embodiment of the present invention. The left part of figure 1 is a view from above showing a plurality of pairs of plate shaped elements which are protruding at right angles from a surface, i.e. in the y-direction. The pairs are arranged at constant distances D apart in the spanwise direction, i.e. the z-direction. A main flow of fluid flows in the streamwise direction, i.e. the x-direction. All of the elements are arranged at angles β relative to the streamwise direction. The individual elements of each of the pairs are arranged symmetrically relative the streamwise direction and are angled in opposite direction, although at the same angle β relative the streamwise direction, such that the distance between the elements increase in the streamwise direction. The individual elements of each pair are separated an average distance d in the spanwise direction, i.e. the spanwise intra-pair separation is d. The elements are arranged along a straight line in the streamwise direction corresponding to a distance X_MVG from the leading edge of the surface (not shown). The right part of figure 1 is a side view of one of the elements. The element is triangular, i.e. has a height which is linearly increasing from zero at the front edge pi to a maximum height h at the rear edge ph. The front edge pi is arranged upstream the rear edge ph such that the height of the elements is increasing in the streamwise direction. The element has a base length L.

Fig. 3 is a side view illustration of different possible geometries of an element of a miniature vortex generator according to embodiments of the present invention. All embodiments of the element has a base length L. According to a first embodiment illustrated in the figure, the element is triangular, i.e. it has a height which is linearly increasing from zero at the front edge pi to a maximum height h at the rear edge ph. According to a second embodiment illustrated in the figure, the element is of constant height, i.e. the height is h at both the front edge pi and the rear edge ph. The dashed lines in the figure illustrate other embodiments, where the height increases from zero height at the front edge pi to the rear edge ph along different non-decreasing curves, such that the height is greater than or equal to the height of the first embodiment and less than or equal to the height of the second embodiment along the length of the element. In other embodiments, the height is non-decreasing along the length, and non-zero at the front edge pi. In yet other embodiments, the height increases from the front edge pi to the rear edge ph along non- decreasing curves such that the height is less than the height of the first embodiment along a portion or the entire length of the element. The height of the elements should preferrably be non-decreasing in order to achieve the desired streaky base flow which appears to be related to the effect of delaying transition to turbulence. hU

Fig. 4 shows a diagram illustrating the upper and lower limits of — - which

υ

the maximum height h of the elements of a miniature vortex generator according to an embodiment of the invention fulfils. x U

Fig. 5 shows a diagram illustrating the upper and lower limits of ^ ^∞ which

υ

the distance X_MVG from the leading edge of the surface at which each of the elements of an embodiment of the invention is arranged fulfils.

Figure 6 shows measurement results comparing boundary layer energy with and without a miniature vortex generator. The figure shows disturbance energy E in the boundary layer as a function of the position downstream of the MVG, where is the streamwise disturbance amplitude.

The curve marked with circle symbols shows the energy distribution without use of a MVG, and the square marked curve shows the energy distribution with a MVG. The measurements are performed in a wind tunnel with a non- dimensional disturbance frequency F=175, where = ^27r^ ^v · 10⁶ , and v is the kinematic viscosity of the fluid, and f is the disturbance frequency. The height h of the elements of the MVG correspond to Re_h=452, where Re_h is the

Reynolds number above the surface at the location of the MVG height defined h ' ti( v — IT )

by Re_h = — - . From figure 6 it is noted that the disturbance energy is v

damped by a factor of approximately 10000 when using a MVG compared to without a MVG. A simple estimation of the skin-friction drag reduction based on standard empirical relations of laminar and turbulent boundary layers, only considering the interval of x from 1200 mm to 1700 mm, amounts to a value of approximately 20%.

Figures 7a and 7b show measurement results illustrating the effect of miniature vortex generators on the disturbance growth in the boundary layer. The initial disturbance has lower amplitude compared to the amplitude in the TS wave it excites and which is seen in the data shown in figure 6.

Disturbance growth is quantified as an integral measure (A^yz _Ts) of the amplitude of the TS waves integrated over the yz-plane (i.e. the plane defined by the surface normal and the spanwise direction). The figures show the logarithm of this integral measure normalized by the integral measure at a reference position (A^yz _Ts_,o) in the streamwise direction as a function of the streamwise position. The miniature vortex generators (MVGs) used for the measurements in fig. 7a-b comprise triangular elements.

Figure 7a shows the results of experiments at a constant free-stream velocity. The curve marked with star symbols (*) shows the disturbance growth without an MVG. In figure 7a, three additional curves are shown with data measured using MVGs having elements with three different heights corresponding to

Re_h=(220, 300, 401 ), where Re_h is the Reynolds number above the surface at h ' ti( v — IT )

the location of the MVG height defined by Re_h = — - , where v is the v

kinematic viscosity of the fluid. The arrows in the figure point toward increasing values of Re_h. From figure 7a it can be seen that the disturbance is increased a short distance downstream of the reference point when using MVGs. Further downstream however, the disturbance decreases to lower levels when using MVGs compared to without an MVG. The highest attenuation of disturbance is achieved with the MVG having elements with the greatest height (corresponding to Re_h=401 ).

Figure 7b shows the results of experiments at a higher free-stream velocity compared to in figure 7a. The curve marked with star symbols (*) shows the disturbance growth without an MVG. The disturbance growth without an MVG is similar to that shown in fig. 7a, although of a larger amplitude due to the higher velocity. The three additional curves are the results of using MVGs having elements with three different heights corresponding to Re_h=(339, 452, 593). The arrows in the figure point toward increasing values of Re_h. From figure 7b it can be seen that the disturbance is increased a short distance downstream the reference point for all three cases when using MVGs. Unlike the results shown in figure 7a, the disturbance does not decrease further downstream to lower levels than without MVGs for all three cases. In fact, the disturbance actually increases when using the MVG corresponding to

Re_h=593. It can thus be concluded that the presence of higher elements in the MVG is only beneficial up to a certain limit, above which the MVG instead provides an earlier breakdown to turbulence. On the other hand, it can also be concluded from measurements such as the one shown in fig. 7a that if the heights of the elements in the MVG are too low, the stabilizing effect is small, and the MVG will only delay transition delay to turbulence by a small amount. Based on fig. 7a-b, it can be concluded that an optimum height of the elements in the MVG resulting in a high attenuation of the disturbance energy may be found. The height of the elements in the MVG is correlated to the amplitude of the streaky base flow. Elements having greater heights result in a streakier base flow with amplitudes that are larger and which declines more rapidly downstream in the streamwise direction compared to elements having smaller heights. The results shown in fig. 7b, where the disturbance increases (using the MVG corresponding to Re_h=593), correspond to an amplitude of the streaks which is too large to achieve the desired effects. Consequently, in order to achieve the desired effect of delaying transition to turbulence, it is important to choose the height of the elements in the MVG carefully. It should be noted that the upper limit defining the heights of the elements in the MVG is more critical than the lower limit. As discussed above, if the height of the elements in the MVG is too large, i.e. above the limit, the transition to turbulence may be advanced rather than delayed. The lower limit is merely chosen to achieve a MVG which results in a delay of the transition to turbulence significant enough to distinguish the invention from the prior art in terms of the effects achieved. The maximum height h of each of the elements may thus alternately fulfil:

The parameter a is less than or equal to 1 and greater than or equal to 0.1 , such as greater than or equal to 0.2, such as greater than or equal to 0.3, such as greater than or equal to 0.4, such as greater than or equal to 0.5, such as greater than or equal to 0.6, such as greater than or equal to 0.7, such as greater than or equal to 0.8, such as greater than or equal to 0.9.

Figures 8a-b show measurement data indicating the effect on the streak amplitude evolution in the streamwise direction. The measurement data are based on 14 different MVG configurations. Streak amplitude is quantified by an integral measure A^int _st of the streak amplitude of integrated over the yz- plane (i.e. the plane defined by the surface normal and the spanwise direction) according to:

A^ i r l = -J- / ^{+ t "} / " \ ( i x y z - 0 i.t, a) \ άηάζ where

L (x, y. z ) j_S the |_oca| velocity at position (x, y, z) in the boundary layer, and L {x. ) j_S the average velocity. Fig. 8a shows this integral measure as a function of the streamwise position x downstream of the position X_MVG from the leading edge at which the elements in the MVG are arranged. Fig. 8b shows the integral measure A^int _st normalized by the absolute integral amplitude value A^int+ _St as a function of the streamwise position downstream of a non-dimensional streamwise coordinate ξ defined according to:

The constant 6.5x10^"7 is an empirical value which has been fitted to the experimental data shown in fig. 8a-b. Consequently, it may need to be adjusted if for example another surface, having different roughness

properties, is used. From fig. 8a-b it can be seen that the streak amplitudes for all experiments achieves maxima certain distances downstream of the MVGs. From fig. 8b it can also be noted that these maxima occur at This allows the normalized streak amplitude integral measure to be expressed as a function of ^according to:

A int i \ iint+ ξ

st

Based on this expression, it is possible to re-scale the MVG design

parameters depending on the conditions of the flow. Given the results above, it can be concluded that the streak amplitude, and thereby the effect of delaying transition to turbulence depends on the position X_MVG from the leading edge at which the elements in the MVG are arranged. It is thus important to choose the position X_MVG carefully in order to achieve the desired effect.

The experimental setup resulting in the measurements shown in figures 7a-b and 8a-b, and also resulting in the expermiental data which the present invention is based on, consists of a flat plate having a leading edge upstream which a flow is introduced in a streamwise direction. In a first region downstream the leading edge, a 2D laminar boundary layer develops on the flat plate, while TS waves of small amplitude are generated in a second region further downstream by means of blowing and suction through a spanwise slot in the plate. The unsteady blowing and suction may be created for example by means of a sealed loudspeaker connected to the slot. The TS waves of small initial amplitude grow into relatively larger amplitude waves in the downstream direction. In a third region even further downstream, a 3D streaky base flow is generated by a MVG. In a fourth region, yet further downstream, the amplitude of the streaky base flow has finally decayed and the 2D base flow found in the first region will eventually be recovered, The upper and lower limits as defined by the present invention have been achieved by performing such experiments at varying free-stream velocities, varying heights and positions of the elements of the MVG and over varying surfaces, All experiments have been performed with MVGs with triangular elements. Similar results are expected for MVGs having elements with other geometries with non-decreasing heights. Performing the corresponding experiments for other geometries in order to determine the upper and lower limits defining the MVGs lies within the abilities of the person skilled in the art.

In one series of experiments where positive results were achieved with respect to delay of transition to turbulence, the elements had maximum heights in the range from 1 .1 to 1 .5 mm, lengths of 3.25 mm, intra-pair separations of 3.25 mm and thickness of 0.3 mm. The boundary layer thickness was in the range from 4 to 9 mm.

Although exemplary embodiments of the present invention have been shown and described, it will be apparent to the person skilled in the art that a number of changes and modifications, or alterations of the invention as described herein may be made. Thus, it is to be understood that the above description of the invention and the accompanying drawing is to be regarded as a non- limiting example thereof and that the scope of the invention is defined in the appended patent claims.

Claims

1 . A miniature vortex generator for delaying transition to turbulence in a flow of a fluid over a surface in vicinity of a leading edge, wherein said fluid has a kinematic viscosity v and said flow has a free-stream velocity U_∞ , wherein said free-stream velocity varies in a streamwise direction of said flow approximately as a power function of the distance from the leading edge with a leading power m, said generator comprising an array of substantially plate shaped elements protruding from said surface, wherein each of said elements is arranged at a distance from said leading edge in said streamwise direction, and wherein each of said elements is arranged at an angle in the range from 9 to 18 or from 6 to 15 degrees relative to said streamwise direction, and wherein the height of each of said elements is non-decreasing with respect to said streamwise direction, and wherein the maximum height h of each of said elements fulfils:

2. The miniature vortex generator according to claim 1 , wherein each of said elements is arranged a distance X_MVG from the leading edge fulfilling: m

Lower limit of ^{Xmvg U∞} Upper limit of ^{Xmvg U∞}

V V

-0.09 2370 9130

-0.08 3850 16500

-0.05 9230 39500

0 24100 103000

0.09 78000 334000

0.2 210000 897000

0.5 1130000 4850000

3. The miniature vortex generator according to any one of the preceding claims, wherein each of said elements has a length in the range of two times to three times the maximum height of said element.

4. The miniature vortex generator according to any one of the preceding claims, wherein the spanwise separations of said elements lie in the range from 5 to 17 times the height of said elements.

5. The miniature vortex generator according to any one of the preceding claims, wherein said elements are arranged in pairs and symmetrically with respect to said streamwise direction.

6. The miniature vortex generator according to claim 5, wherein the spanwise intra-pair separations d of the elements of said pairs lie in the range from 0.75 to 1 .25 times the length of said elements.

7. The miniature vortex generator according to claim 6, wherein the spanwise inter-pair separations D of the pairs lie in the range from 3.5 to 4.5 times said spanwise intra-pair separations d.

8. The miniature vortex generator according to any one of the preceding claims, wherein the thickness of each of said substantially plate shaped elements is less than the height of said element.

9. The miniature vortex generator according to any one of the preceding claims, wherein the heights of said elements are linearly increasing with respect to said streamwise direction.

10. The miniature vortex generator according to any one of the preceding claims, wherein said elements have constant heights.

1 1 . The miniature vortex generator according to any one of the preceding claims, further comprising a second array of substantially plate shaped elements, wherein said elements of said second array are arranged downstream the elements of said array.

12. The miniature vortex generator according to claim 6, wherein the spanwise inter-pair separations D of the pairs lie in the range from 3.0 to 4.5 times said spanwise intra-pair separations d.

13. The miniature vortex generator according to claim 1 , wherein each of said elements is arranged at an angle in the range from 9 to 18 degrees relative to said streamwise direction.

14. The miniature vortex generator according to claim 1 , wherein each of said elements is arranged at an angle in the range from 7 to 1 1 degrees relative to said streamwise direction, such as in the range from 8 to 10 degrees relative to said streamwise direction, such as about 9 degrees relative to said streamwise direction.

15. Use of an array of substantially plate shaped elements protruding from a surface in vicinity of a leading edge of said surface for delaying transition to turbulence of a flow of fluid over said surface, wherein said fluid has a kinematic viscosity v and said flow has a free-stream velocity U_∞ , wherein said free-stream velocity varies in a streamwise direction of said flow as a power function of the distance from the leading edge with a leading power m, and wherein each of said elements is arranged at a distance from said leading edge in a streamwise direction of said flow, and wherein each of said elements is arranged at an angle in the range from 9 to 18 or from 6 to 15 degrees relative to said streamwise direction, and wherein the height of each of said elements is non-decreasing with respect to said streamwise direction, and wherein the maximum height h of each of said elements fulfils: m hU hU

Lower limit of —— Upper limit of ——

V V

-0.09 264 565

-0.08 274 600

-0.05 308 654

0 347 727

0.09 404 851

0.2 459 967

0.5 578 1206