Recirculation regions in wakes with base bleed

K. Steiros,1, ∗ N. Bempedelis,1, 2 and L. Ding3

1Department of Aeronautics, Imperial College London, London SW7 2AZ, UK2Department of Mechanical Engineering, University College London, London WC1E 7JE, UK

3Department of Mechanical and Aerospace Engineering,Princeton University, Princeton, NJ 08544, USA

(Dated: January 6, 2021)

The appearance of detached recirculation regions in wakes with base bleed determines the aero-dynamic properties of many natural organisms and technological applications. In this work weintroduce an analytical model which captures certain key dimensions of the recirculation regionin the wake of porous plates of infinite aspect ratio, along with the porosity range over which itexists, when vortex shedding is absent or suppressed. The model is used to interpret why the re-circulation region (i) emerges, (ii) migrates away from the body with increasing base bleed, (iii)disappears at a critical bleed and (iv) is partially insensitive to variations in the Reynolds number.The model predictions show considerable agreement with data from laboratory experiments andnumerical simulations.

I. INTRODUCTION

It is well known that the near wake of solid bluff bodies is characterized by an attached recirculation region, whichdetermines much of the aerodynamic properties of the configuration [1]. An interesting phenomenon occurs whenfluid bleeding is superimposed on the wake, i.e. when the body is permeable, or when fluid is actively injected at itsbase; the recirculation region detaches, and, as fluid bleeding is increased, moves away from the body rear. Whenbleeding exceeds a critical value, the region suddenly breaks down and disappears [2–8].

This behaviour has profound effects on the bluff-body drag and stability of the flow [8, 9]. It thus comes as nosurprise that several natural organisms have evolved to possess porous membranes [10] and wings [11], which controlthe recirculation region and contribute to a more stable and efficient flight. In engineering, base bleed is taken intoaccount when designing efficient airfoils [2, 12], dynamic mixers [13], or when modelling wind turbines and wind farms[14, 15], to name a few applications.

Despite the above, the reasons behind the emergence, evolution and disappearance of the detached recirculationregion are not clear. The recirculation can be expected to be connected to the intense vorticity of the shear layerswhich induce backflow velocities in the wake [9]. However, the vorticity magnitude alone is insufficient to act as acausal explanation for the state the flow system settles on. As a matter of fact, several potential flow bluff-bodymodels can be found in the literature, the classical model of Kirchhoff [16] being one example, which predict largevalues of shear layer vorticity, but do not include a recirculation bubble in their predictions (for extensions of theKirchhoff model with a vortical wake see [17, 18]). Therefore, apart from the magnitude of vorticity, an explanationof the recirculation region should include at least some information on the distribution of this vorticity, which in turncan be thought to be a function of the diffusion process of the flow (turbulent or molecular) and various constraints(e.g. momentum and mass conservation). This fact was pointed out in the early investigations of Bearman [12] andLeal and Acrivos [2] who postulated that the recirculation region is a product of the entrainment process of the freeshear layers.

In this paper, we use a combination of potential flow modelling, experiments and numerical simulations, and attempta qualitative explanation on why the recirculation emerges, migrates away from the porous body and disappears. Toperform that, we consider the simplest wake with base bleed, that of an infinite-aspect-ratio porous plate, and createa theoretical model which, despite its simplicity, is able to capture, qualitatively, certain basic dimensions and trendsof the bubble. The analysis is strictly valid only for the special case of steady wakes, but our arguments, togetherwith the offered insights, can be thought to also be relevant to the more complex and/or unsteady configurations thatappear in nature and engineering.

The structure of the article is as follows. Section II introduces the phenomenology of the problem, and discussessome major assumptions of the ensuing analysis. Section III presents the novel analytical model for the porousplate recirculation region, and compares its predictions with experimental and numerical data. Finally, section IVsummarizes the results, and discusses the extensions of this study.

∗ k.steiros13@imperial.ac.uk

arX

iv:2

008.

0835

6v3

[ph

ysic

s.fl

u-dy

n] 5

Jan

202

1

2

FIG. 1. Steady wake bubble boundary and slipstream of plates of variable porosity, β, as measured using PIV at Re = 6× 103.The colorbar corresponds to the time-averaged non-dimensional streamwise velocity. X is the bubble detachment length andh/2 its maximum half-height. β is the open area ratio of the plate. The dashed line signifies the presence of a splitter plate,which was used to stabilize the wake in the cases β = 0, 0.1 and 0.2. For β = 0.3 vortex shedding is already suppressed by thelarge base bleeding [3], rendering the splitter plate superfluous.

II. BACKGROUND AND PRELIMINARY CONSIDERATIONS

Figure 1 shows the recirculation bubble of four plates of different porosities for the fully turbulent regime (Re =6× 103 based on the plate height d and free-stream velocity U∞), measured using Particle Image Velocimetry (PIV)(the details of the experimental set-up are provided in appendix A 1). The depicted flow fields correspond to “steady”wakes, where vortex shedding is suppressed, either with the use of a splitter plate (cases β = 0, 0.1, 0.2, where β isthe open area ratio defined as the porous plate area over the gross plate area), or from the large base bleeding (caseβ = 0.3), which acts as an effective splitter plate [3]. We note that for the case β = 0.3 the bubble streamlines didnot reconverge to a stagnation point (possibly due to 3D effects), and for this reason we chose to depict the bubbleas open. We also note that regardless of the stabilization technique (splitter plate or wake bleeding), the shear layerswill eventually become unstable if allowed to interact, and will generate vortex packets. However, these will be muchweaker and will be formed far downstream of the body, compared to the the case where no stabilization is used, sothat their effect in the near wake can be thought to be negligible (see discussion in Steiros et al. [19]).

Similar to past investigations [3], it is observed that as base bleeding increases, the recirculation bubble detachesfrom the plate and migrates away from the body, shrinking in size, until it abruptly disappears at a critical openarea ratio between β = 0.3 and β = 0.4 (no recirculation bubble was observed at β = 0.4). An analogous behaviouris observed in the laminar regime [2], with the difference that the detachment length X increases linearly with Re,whereas in turbulent conditions the bubble dimensions are generally thought to be Re-independent. On the otherhand there are indications that the bubble height is Re-independent even in the laminar regime [2].

Our aim is to provide an explanation for the above observations, and create a model which can capture, qualitatively,the dimensions X and h, and the porosity range where the bubble exists. Our starting point is the porous plate modelof Steiros and Hultmark [20], which is based on potential flow theory, “corrected” using the constraints of mass andmomentum conservation. This model does not predict nor presupposes a recirculation bubble and shear stresses inthe wake. Despite that, it yields accurate predictions of the drag of infinite aspect ratio porous plates, as long asvortex shedding is absent or suppressed. This model also yields accurate predictions for the wake pressure, pw, atleast for the case where β = 0. Moreover, the external flow velocities (i.e. outside the wake) have been shown [21] tobe described reasonably well by the potential flow description that Steiros and Hultmark [20] use, which is identicalto the one presented in Taylor [22] when considering the outer flow.

Given the above, we make the assumption that inclusion of the recirculation bubble and shear stresses in theanalysis will only alter the wake velocities, and will not affect the plate drag, wake pressure, and wake geometrythat are predicted by the modified potential flow model of Steiros and Hultmark [20]. The prediction of Steiros andHultmark [20] for the wake velocity after the initial wake expansion, Uw (see figure 2), is given below as it is usedrepeatedly in the analysis that follows:

UwU∞

=u∗

2− u∗, (1)

3

p8

U8

pw

Uwu

y

x

slipstream

H

d

FIG. 2. Potential flow representation of the steady wake by Steiros and Hultmark [20]. U∞ and p∞ denote the free-streamvelocity and ambient pressure respectively. Uw and pw denote the wake velocity and pressure after the initial wake expansion.H is the wake height after the initial expansion, and d is the plate height. u is the velocity immediately upstream/downstreamof the plate.

where u∗ = u/U∞ is the plate bypass ratio (see figure 2 for a definition of the flow quantities). This variable is centralto the model of Steiros and Hultmark [20], and is the independent variable of the models developed in the presentstudy. However, in most cases, this is not a readily available quantity. A more straightforward approach would be toexpress all quantities as a function of the open area ratio β, and for that an expression linking β with u∗ is needed.Taylor and Davies [23] have provided and validated such an equation when friction losses are negligible. This, togetherwith the drag expression of Steiros and Hultmark [20], yields the following expression (see appendix B for more detailson the derivation)

β ≈√

3(2− u∗)u∗√−3u∗3 + 2u∗2 − 4u∗ + 8

. (2)

Note that friction losses can be thought to be negligible when the flow is fully turbulent, or when the plate isvanishingly thin. In that case, the effects of other properties, such as the permeability, can be thought to be secondary(see for instance [24] and [25]). In the remainder of the text, the above equation will be used to plot the predictionsas a function of β instead of u∗.

III. CHARACTERISTICS OF THE RECIRCULATION BUBBLE

A. Bubble height and disappearance

We assume the flow field shown in figure 3, where a recirculation bubble is superimposed on the potential flowmodel depicted in figure 2. The proposed flow model can be realistic only if it is consistent with conservation of massand momentum. We thus require that these constraints be fulfilled in control volume C1, which is positioned at alocation where the bubble is sufficiently parallel to the slipstream, so that the characteristic (or y−averaged) flowvelocity U2 between the bubble and the slipstream can be assumed to be approximately constant with the streamwisedistance x. Figure 1 shows that this condition is locally satisfied around the location where the bubble assumes itsmaximum height.

For sufficiently high values of Re, the shear stress on the slipstream or the bubble streamlines is mainly the Reynoldsstress ρu′v′. Following Roshko [1], this can be modelled as Ctρ(Ua − Ub)2, where Ua and Ub are the characteristicvelocities at the two sides of the shear layer, and Ct is a coefficient of proportionality which is assumed constant at afirst approximation. Application of momentum conservation on control volume C1 and on the bubble yields

2Ctρ(U∞ − U2)2δX = δPH ,

2CtρU22 δX = δPh ,

(3)

where δX is the length of control volume C1, δP is the pressure increase inside the control volume C1 and the factortwo is due to the contribution of both (upper and lower) shear layers. In the above it was assumed that the velocity

4

p8

U8

uy

xH

d U2 C1

X

h

slipstream

H8

Pσ C2

δX

FIG. 3. Extension of the potential flow representation of the wake of figure 2, by addition of a recirculation region. The dashedregions C1 and C2 denote control volumes.

outside the wake is U∞, and zero inside the bubble, the latter following Roshko [1]. These velocity values are notstrictly correct (see figure 1) and should only be considered as characteristic of the corresponding mixing layers.Different velocity values to the ones assumed would alter, slightly, the predicted height. The predicted trends andgeneral arguments would, however, remain unaltered.

If the bubble were not present, the streamwise velocity at C1 would be predicted from the potential flow model ofSteiros and Hultmark [20] to be Uw, i.e. the velocity of the expanded wake, given by equation (1). With the additionof the bubble in the analysis, the cross-sectional area that the fluid may pass through is reduced, and the new velocity,U2, can be calculated using mass conservation as

ρU2(H − h) = ρUwH . (4)

Combination of equations (1), (3) and (4) yields

h

H=

(u∗(

1− hH

)(2− u∗)− u∗

)2

, (5)

which links the non-dimensional maximum bubble height h/H (i.e. at the location where the bubble is parallel tothe slipstream) with the bypass ratio u∗ or, equivalently, with the open area ratio β. The only physically relevantsolution to the above equation is

h

H=

3u∗ − 2−√

(u∗ − 2)(5u∗ − 2)

2(u∗ − 2). (6)

The predictions of equation (6) are plotted in figure 4(a), together with values measured from the PIV experimentsand Large-Eddy Simulations (see appendix A for details). The observations are in qualitative agreement with thetheoretical predictions. When 0.4 < u∗ ≤ 1, equation (6) does not yield real values: the model-bubble is incompatiblewith the constraints imposed by momentum and mass conservation. More specifically, the system (3) shows that theexternal shear generates a pressure gradient in the wake, and thus constrains the bubble shear, as the latter must alsoproduce the same pressure gradient. The above can only occur when the bubble has a specific height as this affectson one hand the shear (by changing the velocity between the slipstream and the bubble U2) and on the other thepressure force on the bubble. Above a critical porosity value no bubble is compatible with the above constraints, and itmust therefore disappear. Our simplified model predicts this disappearance to occur at β ≈ 0.34 (which is equivalentto u∗ = 0.4, according to expression (2)). In agreement with that prediction, our measurements indicate that the“critical” porosity β lies somewhere between 0.3 and 0.4, while our simulations predict a critical porosity of β = 0.35,where the recirculation bubble appears intermittently in time, and is absent when considering the time-averaged flowfield.

It must be mentioned that when β → 0 (and h→ H), there is not enough separation between the bubble boundaryand the slipstream to consider them as two separate shear layers (see figure 3). In that case, the system of equations(3), and thus the above analysis, is not valid. However, equation (6) yields the correct prediction of h/H = 1 for

5

(a) (b)

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

β

h/H

PIV

LES

Model

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

β

X/d

PIV

LES

Castro

Model

FIG. 4. Normalized (a) bubble height and (b) detachment length, as a function of β. In the dashed part of the prediction, whereβ → 0, the theoretical analysis is not strictly valid. The hatched areas represent the region where the mass and momentumconservation laws cannot be fulfilled. The dashed vertical line indicates the porosity where the recirculation bubble disappears,according to the numerical simulations. Where available, results from the experiments of Castro [3] are also included. Theratio h/H was measured at the location of maximum bubble height, while the distance X is the minimum distance betweenthe bubble and the plate.

β = 0, while we expect the region of invalidity to be small, as for β = 0.1 the separation of the bubble and slipstreamis already of significant extent (see figure 1). We thus use equation (6) even when β → 0, and we expect that thedeviation from the “correct” prediction is small. To emphasize the above, the part of the prediction where we expectit to not be strictly valid has been marked with a dashed line in figure 4(a).

B. Bubble detachment and emergence

We now consider control volume C2 (see figure 3), which is bounded by the slipstream, and extends from farupstream of the plate to the tip of the bubble, i.e. at distance X from the plate. As argued in section II, we assumethat the pressure at this location is equal to pw, i.e. the wake pressure as predicted by Steiros and Hultmark [20]. Ifthe bubble and shear stresses were absent from the analysis, the problem would be reduced to the one of Steiros andHultmark [20], and in that case, a momentum balance in C2 would yield

D = ρud(U∞ − Uw) + p∞H∞ − pwH + Pσ , (7)

where D is the drag and Pσ is the pressure force on the slipstream. Steiros and Hultmark [20] employed a slightlydifferent control volume, but under identical assumptions, and their results are thus equivalent.

We now increase the complexity of the analysis by adding a bubble and shear stress on the slipstream. As arguedin section II, it is assumed that only the wake velocities are altered from the added complexity. Thus, a momentumbalance in C2 now yields

D = ρud(U∞ − U2) + p∞H∞ − pwH + Pσ + Fs . (8)

For high Re, the forcing on the slipstream Fs is mainly the Reynolds stress. Similar to the discussion in sectionIII A, this is modelled as being proportional to the square of the difference of the characteristic velocities across theslipstream. An approximation is to consider the outside velocity to be U∞, and the inside one to be u. Thus, theforce due to shear stress becomes

Fs ≈ 2ρC ′t(U∞ − u)2X , (9)

6

where the factor 2 is due to the two (upper and lower) shear layers, and C ′t an empirical parameter (provided fromexperiments or numerical simulations). Combining equations (7), (8) and (9) we obtain

2ρC ′t(U∞ − u)2X ≈ ρud(U2 − Uw) . (10)

We interpret the above equation by following an argument analogous to that of Bearman [12] and Leal and Acrivos[2]: immediately downstream of the plate, the shear on the slipstream will cause an acceleration of the wake fluidwhich is closest to the slipstream and, in order for mass to be conserved, the fluid at the core of the wake will haveto be continually decelerated. When it reaches zero velocity, a recirculation bubble will form. At this exact location,the momentum flux will be approximately ρudU2, i.e. increased by ρud(U2 − Uw) compared with the case where abubble is not included in the flow description. Our simplified analysis assumes that this increase in momentum fluxis exactly provided by the shear stress on the slipstream.

Combination of equations (1), (4), (6) and (10), leads to the following expression for the normalized detachmentlength X/d

X

d=

1

2C ′t

u∗2

(1− u∗)2(2− u∗)3u∗ − 2−

√(u∗ − 2)(5u∗ − 2))√

(u∗ − 2)(5u∗ − 2))− u∗ − 2, (11)

where C ′t must be determined from the shear stress on the part of the slipstream which extends from the plate tothe start of the bubble. As described in appendix A, by integrating the shear stresses on the slipstream up until thebubble, and then averaging across all tested cases, we obtain a value of C ′t = 0.0257 for our laboratory, and C ′t = 0.0395for our numerical experiments. These values are somewhat different, but we note that our region of integration liesvery close to the plate, where the flow is extremely inhomogeneous and sensitive to boundary conditions, which areslightly different in the simulations and experiments. At larger downstream distances the initial conditions (e.g. shapeof holes) are expected to not affect the statistics, as shown in the comparison between experimental and numericalresults (see appendix A 2).

In figure 4(b) we plot equation (11) using the above two values of C ′t to form an envelope. We observe that thepredictions are in qualitative agreement with the computed/measured values of X/d, suggesting that the argumentsof the model are not far from what occurs in reality.

C. Laminar regime

The analysis presented in the above sections concerns the fully turbulent regime, where the bubble dimensions areexpected to be independent of the Reynolds number, in accordance with equations (6) and (11). In the laminar regimeon the other hand, Leal and Acrivos [2] observed that while the normalized height of the centre of the bubble remainsRe-independent, the detachment length X increases linearly with Re. We will attemt to explain these observationsusing the proposed framework.

We consider a steady laminar wake and assume that the present analysis is capable of describing the flow field. Atfirst this seems counter-intuitive, given that one would expect that an analysis based on potential flow theory is likelyto be valid only for very high Reynolds numbers. Nevertheless, the arguments and results of Acrivos et al. [26] andLeal and Acrivos [2] indicate that an “infinite” Re analysis is appropriate for bluff bodies whose Reynolds numbersare as low as 25.

The analysis presented in sections III A and III B may thus be repeated, with the only difference being thatthe stresses on the slipstream and bubble are now predominantly viscous rather than turbulent. These can beapproximately modelled using a standard boundary layer analysis with l as the characteristic length

Fs = CvµUa − Ub

l

√Rel , (12)

where µ is the dynamic viscosity of the fluid, (Ua − Ub) is the velocity difference across the shear layer and Cv isa coefficient of proportionality. A momentum budget similar to the one presented in section III A now yields thefollowing system of equations

7

(a) (b)

50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2

Re

h/H

50 100 150 2000

0.5

1

1.5

2

Re

X/d

FIG. 5. Normalized (a) maximum bubble height and (b) detachment length, as a function of Re in the laminar regime, for athin plate where β = 20.6%. The ratio h/H was measured at the location of maximum bubble height, and the distance X asthe minimum distance between the bubble and the plate. Symbols: DNS. Line: linear fit.

2CvµU∞ − U2

l1Re−1/2l1

δX = δPH ,

2CvµU2

l2Re−1/2l2

δX = δPh .

(13)

where l1 and l2 are the boundary-layer characteristic lengths for the slipstream and bubble, respectively.Calculation of the normalized bubble height h/H requires taking the ratio of the above equations, and thus any

dependency on viscosity cancels out, as per the observations of Leal and Acrivos [2].With respect to the detachment length, the laminar equivalent of equation (10) is

2Cvν(U∞ − u)

√U∞X

ν= ud(U2 − Uw) , (14)

where ν = µ/ρ and the characteristic length is taken equal to the detachment distance X. This can be rearranged toyield

X

d= Ref(u∗) , (15)

which predicts a linear increase of the detachment length with Re in the laminar regime, as per the observations ofLeal and Acrivos [2].

In figure 5 we examine the validity of the predicted trends by comparing them with data from a series of DirectNumerical Simulations (DNS) of a plate of β = 20.6%, for 40 ≤ Re ≤ 200 (see appendix A for numerical details). Thesimulated plate was very thin (≈ 1.77% of the plate height d), so as to minimize viscous effects due to the frictioninside the holes, which can be significant in the laminar regime, and may thus induce effects which are not taken intoaccount in the analysis. In accordance with our theoretical predictions, the normalized bubble height is independentof the Reynolds number, while the detachment length grows linearly with Re.

IV. CONCLUDING DISCUSSION

This work introduces a model for the prediction of the height and detachment length of the recirculation bubblethat appears in the steady wake of porous plates when their open area ratio is relatively small. The model extends the

8

FIG. 6. Time-averaged recirculation bubble for different values of porosity β, as measured using PIV (see appendix A 1), atRe = 6 × 103. The colours correspond to the time-averaged non-dimensional streamwise velocity. Vortex shedding makes thewake unsteady for β = 0, 0.1 and 0.2, but is suppressed for β = 0.3 due to the large bleeding [3]. The bubble evolution isqualitatively similar to the case where the wake is stabilized using a splitter plate (figure 1).

work of Steiros and Hultmark [20] for drag and flow field of porous plates, by superimposing a recirculation bubble ontheir predicted flow field. The main assumptions are that (i) the flow field is steady and two-dimensional and can bewell-represented using potential flow physics; (ii) the addition of stresses (turbulent or viscous) on the wake slipstreammodify the potential flow description only by altering the velocities inside the wake, leaving the pressure, plate forcesand outer flow unaffected; (iii) the characteristic velocity inside the bubble is zero; (iv) the bubble boundary is parallelto the slipstream at the position of maximum bubble height; and that (v) the slipstream stresses can be modelled ascanonical free shear layers (turbulent stresses) or laminar boundary layers (viscous stresses).

On the basis of the above theoretical analysis, we attempt interpretations on why the bubble (i) emerges, (ii) movesdownstream with increasing base bleed and (iii) disappears at a critical value of porosity. Moreover, by applying thedeveloped framework in both laminar and turbulent regimes, we predict that the normalized height of the bubbleis always Re-independent, while the detachment region grows linearly with Re in the laminar regime, but is Re-independent in the turbulent regime, in accordance with past and present observations.

The above analysis concerns the steady wake of infinite aspect ratio porous plates. Nevertheless, the physicalarguments that are presented in this study, along with the various explanations and interpretations, are general, andcan be thought to be applicable to the detached recirculation regions that characterize many other wakes with basebleed that appear in nature and engineering. A pertinent example is those wakes in which bleeding is generated byactively injecting fluid in the wake, rather than passively perforating the body. Such apparatuses are relevant, forinstance, to bluff airfoil flow control [2, 12]. In such cases, a recirculation bubble again appears, which shares allthe characteristics and trends of the recirculation bubble of porous plates, albeit possessing different characteristicdimensions. Another example is the wake of various flowers and seeds. Despite their complexity, these organismsgenerate wakes and detached recirculation regions which are remarkably similar to those of porous plates [10].

When the wake becomes unsteady due to the emergence of vortex shedding, the time-averaged flow picture exhibitsa recirculation bubble, whose trends and physics show some similarity to the ones of the steady case (see figure 6).It is conceivable, then, that the arguments of the present study could still be applicable in a qualitative sense, eventhough the actual wake dimensions cannot be accurately described by the theory presented here, which concernssteady wakes.

ACKNOWLEDGMENTS

Part of this work was conceived while the first author was a post-doctoral associate at Princeton University. Weare grateful to Prof. Marcus Hultmark for helpful comments and discussions, and to Prof. Lex Smits for providingaccess to the water channel where the experiments were conducted.

9

(a)

flow

camera mirror

light sheet

force sensor

(b) (c)

15d

flow 12 d3.3

d

d

Measurement domainy

x

6.1 mm 8.9 mm

d = 30 mm

60◦

FIG. 7. (a) Side and (b) top view of the experimental configuration, and (c) plate design. All tested plates have the samehole-pattern, but different hole diameter.

Appendix A: Details of experimental and numerical methods

1. Experimental apparatus and data processing

The validation experiments were conducted in a free-surface recirculating water tunnel. The tunnel had a 0.46 mwide, 0.3 m deep and 2.44 m long acrylic test section, and operated at a constant free-stream velocity of 0.2 ms−1.The experimental set-up is shown in figure 7.

Porous plates with porosities β = 0% (solid), 10%, 20%, . . . , 60% porosity were tested. The plates were 29 cmlong, 3 cm wide and 0.3 cm thick, were made from aluminium and were perforated with circular holes arranged in theway depicted in figure 7. The hole diameter was varied among the values 3.6, 5.1, 6.2, 7.2, 8.1 and 8.8 mm in orderfor different porosities to be achieved. The channel blockage was 6.5% for the solid plate and lower for the porousplates. The Reynolds number, based on the plate height, Re = U∞d/ν, was 6,000 for all plates. In the experiments,the plates were supported from the top and were positioned near the center of the test section. The plates touchedthe floor of the channel at one end, while the other end extended above the water free surface, in order to suppressthree-dimensional effects arising from the plate edges. For porosities 0%, 10% and 20%, a 2 mm thick splitter platewas positioned at the wake of the plate, spanning approximately 10 porous plate heights. This was done to stabilizethe wake by suppressing the vortex shedding which otherwise appears at these porosities [3]. We note that a splitterplate generates additional effects in the flow field, as for instance skin friction, which are not included in the model.However, the PIV data show relative agreement with both the theoretical model and the LES data (symmentry, nosplitter plate, see section A 2), suggesting that these effects can be largely neglected. Utilization of a splitter plate asa validation tool for steady-wake models has been repeatedly employed in past works (see for instance [1, 26]).

Planar PIV was performed to study the evolution of the recirculation bubble. A dual-cavity Nd:YAG laser (LitronNano PIV, 50 mJ pulses) was used to form a light sheet at half depth of the tunnel. A PIV camera (LaVision ImagersCMOS) was situated underneath the tunnel to image the wake through the reflection of a mirror. A 2× teleconverter(Vivitar) was attached to the camera yielding a magnification of 0.16. In the cross-stream direction, the field of view

10

covered the full height of the plate plus another 2.3 plate heights on a single side of the plate. In the streamwisedirection, velocity data were acquired at five stations that spanned over 12 plate heights downstream of the plate.Different downstream stations were accessed by traversing the plate upstream while keeping the camera and mirrorfixed, since the free stream in the middle portion of the test section was homogeneous. A small overlap region betweenthe measurement planes allowed their subsequent stitching, by finding the location where the velocity profiles fromtwo different planes collapsed in an optimal manner.

At each station, 1,000 image pairs were recorded and analyzed to calculate flow statistics. PIV interrogation wasperformed in DaVis 8.3 using a multi-grid, image deformation strategy [27], with a final interrogation spot size of 2mm × 2 mm and 50% window overlap. Velocity vectors were validated with the universal outlier detection method[28].

The bubble boundary was found by locating the streamlines that reversed their direction in the averaged flow field,and then choosing the one that created the largest bubble. It was often the case that two streamlines were stitchedtogether to form the bubble boundary: one for the front change of direction (bubble tip), and one for the rear. Theslipstream was found by locating the streamline that passed from the plate edge. The empirical coefficient C ′t in thediscussion of section III B was found by calculating the normalized turbulent stress u′v′/(U∞ − u)2 on the slipstream,from a location 0.15d downstream of the plate, to the location where the bubble emerged, and then taking its averagevalue (u∗ was calculated using equation (2)). The downstream distance 0.15d was chosen as a compromise betweenstarting the averaging process as close as possible to the plate, and far enough, in order to minimize the noise due toreflections and shadows. For the cases where a steady detached bubble emerged, (i.e. β = 0.1 and 0.2 with splitterplate and β = 0.3 without splitter plate), C ′t was calculated equal to 0.0257 ± 0.002.

2. Numerical simulations

Simulations were performed using the general-purpose finite-volume toolbox OpenFOAM. For the turbulent flowcases, where the Reynolds number based on the plate height and free-stream velocity was Re = 6240, Large-EddySimulations (LES) using the dynamic one-equation subgrid-scale model of Kim and Menon [29] were performed. Forthe simulations in the laminar regime (section III C), the sub-grid scale model was dropped, and Direct NumericalSimulations (DNS) were conducted. The convective terms were discretized with a second-order central differencingscheme. A second-order backward scheme was used for time discretization.

The porosity of the plates was controlled by changing the size of the homogeneously distributed square-shaped holes.The arrangement of the holes mimicked that of the experiments (shown in Fig. 7), but their shape was different tofacilitate the grid generation process. For the simulations in the turbulent regime, the thickness of the plates wassimilar to those in the experiment, ≈ 10% of the plate height d. Five plates were considered, with porosities equalto β(%) = 0, 9.2, 20.6, 28.3, 35. In the laminar flow cases, the plate that was considered was thinner (≈ 1.77% of theplate height d), so as to minimize viscous effects due to the friction inside the plate holes, which can be significant atlow Reynolds numbers.

Communication of the upper and lower layers was inhibited by simulating only half the plate and enforcing symmetryconditions along the centreline. Vortex shedding was therefore suppressed, in a similar way as when a splitter plate isused. The computational domain consisted of a rectangular box extending 27.5×8×1.7 times the plate height d in thestreamwise, normal, and spanwise directions, respectively (see figure 8). Laminar inflow was imposed at the upstreamboundary, whereas periodic boundary conditions were applied in the spanwise direction. The spanwise extent of thedomain was taken such that precisely five holes could be accommodated along this direction (see figure 8). Thedomain was discretized with ' 3.0× 106 elements in the turbulent flow cases, and ' 1.8× 106 in the laminar ones. Inall cases, 50 cells were used along the span. Data were averaged over a non-dimensional time period correspondingto t∗ = tU∞/d = 104 flow times. For the cases where a steady detached bubble emerged, (i.e. β = 0.092, 0.206 and0.283), C ′t was calculated as described in appendix A 1, and was equal to 0.0395 ± 0.0105.

Validation of the numerical simulations was performed against the PIV measurements (section A 1), for the casewhere β = 30% (the plate porosity in the simulations was β = 28.3%) and no splitter plate is present in the wake.The symmetry condition was removed, and the domain was doubled in size in the normal direction, so that the entireplate could be accommodated. This change mostly affected the flow downstream of the bubble, whereas the keydimensions of the bubble that are discussed in the main body of the paper were largely unaffected. The results fromthe laboratory and numerical experiments are compared in figure 9; good agreement between the two is observed.

Appendix B: Derivation of equation (2)

Steiros and Hultmark [20] provide the following expression for the drag coefficient of an infinite-aspect-ratio plate

11

slip

symmetry

inlet

outlet

periodic

FIG. 8. Computational set-up for simulations of the flow past porous plates (domain and plate not drawn to scale).

(a) (b) (c)

0 2 4 6 8 10 12−0.2

0

0.2

0.4

0.6

0.8

1

x/d

u/U

∞

0 0.5 1 1.5 2−4

−2

0

2

4

6·10−2

y/d

u′ iu′ j/U

2 ∞

0 2 4 6 8 10 12−0.2

0

0.2

0.4

0.6

0.8

1

x/d

|u|/U∞

−4

−2

0

2

4

6·10−2

u′ v′ /U

2 ∞

FIG. 9. (a) Streamwise velocity along wake centreline. (b) Reynolds stresses at a cut in the wake at x/d = 5. Black: u′u′/U2∞,

red: u′v′/U2∞, blue: v′v′/U2

∞. (c) Black: velocity magnitude and red: Reynolds stress (u′v′/U2∞) along the slipstream. Time-

averaged results. Dashed lines: PIV measurements (from a single plane), solid lines: numerical simulations (averaged over thespanwise extent of the domain).

CD =4

3

(1− u∗)(2 + u∗)

2− u∗. (B1)

The above expression provides good agreement with experimental measurements when the wake is steady (i.e. vortexshedding is absent) and the regime fully turbulent.

On the other hand, Taylor and Davies [23] propose the following expression, linking β to u∗, when friction lossesare negligible (so that the effect of permeability can be thought secondary)

CD ≈ u∗2(

1

β2− 1

). (B2)

The above formula has been repeatedly used in ensuing studies of porous plates which consider both 3D (finiteaspect ratio) and 2D (infinite aspect ratio) geometries (see for instance [3, 21]). Note that the critical assumptions ofthe above equation are that the surplus kinetic energy, due to acceleration of the fluid which enters the plate pores,becomes irreversibly heat, due to the expansion at the end of the pores, and that the effect of the vena contracta isnegligible. Taylor and Davies [23] provide experimental validation for the above formula for a variety of porous plates,in fully turbulent conditions (see also Steiros and Hultmark [20] for a reproduction of their validation plot).

12

Combination of the above two equations yields equation (2).

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