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Source models of flow through and around screens and gauzes
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Ocean Engineering 33 (2006) 1884–1895 www.elsevier.com/locate/oceaneng
Source models of flow through and around screens and gauzes F.G. O’Neill FRS Marine Laboratory, 375 Victoria Road, Aberdeen, AB11 9DB, Scotland Received 14 February 2005; accepted 25 October 2005 Available online 19 January 2006
Abstract The source panel method is used to investigate the singularity models of flow through and around screens that have been developed by various authors. These studies are extended to complete a framework of models in terms of the level of complexity, the nature of the flow in the wake and the form of pressure drop boundary condition. The models that best represent the limited experimental data that is available are identified and the need for more experimental data, in order to carry out a thorough assessment, is noted. r 2005 Elsevier Ltd. All rights reserved. Keywords: Source panel method; Permeable boundaries; Screens; Gauzes
1. Introduction Flows through and around screens, gauzes and porous membranes occur in a wide range of engineering applications such as fishing gears and plankton samplers, parachutes and sails, filtration and sifting systems and in wind tunnels to control and modify the velocity field. Laws and Livesey (1978) survey the available literature on the topic of flow through screens and categorise the studies according to whether they were concerned with (i) the characterisation of the flow properties of the screens, (ii) the effect of a screen on the time–mean velocity distributions and (iii) the turbulence distribution downstream of screens. In this paper we consider in greater detail the second of these categories and, in E-mail address: [email protected]. 0029-8018/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.10.009
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particular, examine the singularity models that have been developed to describe the time–mean flow field through and around finite screens in an infinite flow. The first of these was put forward by Taylor (1944) who modelled the flow outside of the wake by considering a screen to be a uniform distribution of centres of resistance and then used Lagally’s theorem to calculate the source strength of an equivalent uniform distribution of sources. Koo and James (1973) ensure that mass and momentum is conserved across the screen and also model the flow in the wake of the screen. By manipulating the stream function to ensure the conservation of mass and momentum across the screen they define a flow that is irrotational everywhere except in the wake region where it is approximately rotational. Cumberbatch (1982) and later Howells and Waechter (1995) employed the hodograph plane and conformal mappings, together with a source distribution along that part of the boundary that represents the screen, to model the flow outside of the wake. Although these models are conceptually similar, insofar as in each case the screen is represented by a singularity distribution, their predictions differ. In this paper we show that these differences are attributable to the relative complexity of the model and assumptions made concerning (i) the nature of the flow in the wake and (ii) the pressure drop boundary condition. In order to eliminate methodological disparities we use the source panel method to model the screen and thus investigate the different assumptions of these authors.
2. The source panel method The source panel method is a well-established technique (especially in the aeronautics industry) for solving potential flow problems. It is particularly suited for modelling flow through permeable surfaces and is essentially the approach used by Taylor, and Koo and James. Here we define a linear element with a constant source distribution to be our basic element. If we consider a distribution of strength m/2p per unit length on the line, g, between points b and c in the complex plane, the complex potential is Z m c F ðzÞ ¼ lnðz gÞjdgj (1) 2p b and the complex velocity is Z dF m c jdgj ¼ . wðzÞ ¼ dz 2p b z g Substituting g ¼ b þ ðc bÞt where t belongs to [0,1] gives Z mjc bj 1 dt , wðzÞ ¼ 2p 0 z b ðc bÞt mjc bj z c ¼ ln 2pðc bÞ zb and with w ¼ u iv, u ¼ RðwÞ and v ¼ IðwÞ.
(2)
ð3Þ
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More specifically, if we consider the complex velocity of a constant source distribution of strength m/2p on the y-axis between points ib and ib, where here b is a positive real number, we have im z ib ln wðzÞ ¼ . (4) 2p z þ ib Substituting z ¼ iy, where y belongs to [0, b] we can show that the velocity at such a panel element is m im by ln , (5) wðiyÞ ¼ þ 2 2p bþy where the sign of the real part (as shown by Taylor, 1944) is such that the normal velocity along one of these linear elements is m/2 in the direction away from the element. With v ¼ IðwÞ, we can show, by straightforward integration that the average tangential speed at a panel element is Z b m by m ln 2 . (6) ln dy ¼ 2pb 0 bþy p Similarly, we can show after substitution and reference tables (Gradshteyn and Ryzhik, 1965) that the root mean square of the tangential speed is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z 1 Z m 1 b 2 by m ln2 q 2 ln dq, dy ¼ 2 2 2p b 0 bþy 2p 0 ð1 þ q Þ m ¼ pffiffiffiffiffi . ð7Þ 12
3. Boundary conditions The drag on a screen is equal to the pressure drop across the screen. The pressure drop is usually expressed in terms of a non-dimensional coefficient k, Dp ¼ 0:5ru2s k,
(8)
where us is the velocity normal to the screen (Taylor, 1944; Koo and James, 1973) (see Fig. 1). A number of authors have identified expressions relating k to solidity and Reynolds number (Brundrett, 1993; Schubauer et al., 1950; Carrothers and Baines, 1975; Hoerner, 1952). However, we will not go into the form of these relationships here. An alternative description of the pressure drop, that has been used by Howells and Waechter (1995) and Cumberbatch (1982), is one based on Darcy’s law where the pressure drop across the mesh is assumed proportional to the flux through the mesh. This can be expressed as Dp ¼ arUus ,
(9)
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vs
vw
U p-∞
1887
uw
us
ps
U∞ p∞
pw
Fig. 1. The s and w subscripts refer to quantities at the free surface of and in the wake of the screen, respectively, while the N and N subscripts refer to quantities upstream and downstream, respectively.
where U is the upstream speed. As for k, a is non-dimensional and will depend on solidity and Reynolds number. (Note for low solidities us tends to U and both these expressions converge for k ¼ 2a.) Bernoulli’s theorem for the steady motion of an inviscid fluid states that p þ 0:5q2 (10) r is constant along a streamline, where p is the dynamic pressure and q is the speed. If we consider the flow around (and through) a screen of finite extent and look at two streamlines, one which goes around the screen, the other which goes through it, we can show that p1 p þ 0:5U 21 ¼ 1 þ 0:5U 21 (11) r r and p1 p þ 0:5U 21 ¼ s þ 0:5q2s , r r
(12)
where the 1 subscript refers to far upstream, the 1 subscript refers to far downstream outside of the screen wake and the s subscript refers to the free surface of the screen.
4. Models of flow through and around screens Here we examine the three different modelling approaches, as determined by the boundary conditions imposed by Taylor, Koo and James, and Cumberbatch. For each approach we consider the cases where the screen is modelled by (i) multiple panel elements, (ii) a single panel element and vs is approximated either by m ln 2/p or pffiffiffiffiffi m= 12 and (iii) a single panel element and vs is assumed negligible. The first of these is the most general. It is equivalent to the model of Koo and James and is the source panel model that best approximates the analysis of Cumberbatch. The third is equivalent to Taylor’s analysis and to an approximation used by Koo and James. The second is an intermediate model and assumes that the tangential velocity can be
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approximated by the mean (in Taylor’s model) or by the root mean square (in the other two). As we see later this choice is informed by examining the drag coefficient in the limit as the screen tends towards impermeability. Furthermore, in each case we investigate both the square and Darcy’s law form of the pressure drop coefficient. 4.1. Taylor model Taylor models the flow outside of the wake by considering the screen to be a uniform distribution of centres of resistance (i.e. a single panel element) and then uses Lagally’s theorem to calculate the source strength of an equivalent uniform distribution of sources. Lagally’s theorem states that a source of strength m/2p at a point in the flow where the velocity is V produces a resistance rVm. For ffia plane pffiffiffiffiffiffiffiffiffiffiffiffiffiffi screen normal to the flow us ¼ U m=2 and the magnitude of V ¼ u2s þ v2s so that at each panel element we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dp ¼ 2rðU us Þ u2s þ v2s . (13) Square law: Equating this resistance to Eq. (8) and non-dimensionalising, we define the multi-panel extension of Taylor’s analysis by ensuring that at each panel qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ku2s ¼ 4ð1 us Þ u2s þ v2s . (14) A simplification of this is to assume that a screen can be modelled by a single panel element. If vs is approximated by its mean value m ln 2/p, we derive the following quartic equation: 2 2 ð1 k2 =16Þp2 4 2p p 3 þ 16 us þ þ 24 u2s 16us þ 4 ¼ 0, 4þ us ln2 2 ln2 2 ln2 2 (15) which can be solved to express us and consequently cd as functions of k. A further simplification, and one which reproduces the analysis of Taylor, is to assume that vs is negligible to give us ¼ 1=ð1 þ k=4Þ. Darcy’s law: If we use Darcy’s law (ie Eq. (9)) we must satisfy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2aus ¼ 4ð1 us Þ u2s þ v2s
(16)
(17)
on each panel element of a multi-panel screen model. As above, if we assume that the screen can be modelled by a single panel and approximate vs with m ln 2/p, we find that at the screen us must satisfy 2 p2 2p þ 16 u3s 4 þ 2 u4s ln 2 ln2 2 ð1 a2 =4Þp2 þ 24 u2s 16us þ 4 ¼ 0. þ ð18Þ ln2 2
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The Darcy law solution analogous to Taylor’s original one, where vs ¼ 0 is us ¼ 1 a=2.
(19)
4.2. Koo and James model Koo and James (1973) propose a formulation that models the flow through, around and in the wake of a screen placed in a parallel-sided channel. By manipulating the stream function to ensure the conservation of mass and momentum across the screen they define a flow that is irrotational everywhere except in the wake region where it is approximately rotational. Here, we apply their theory to the problem of this paper (i.e. a finite screen normal to an infinite flow). In addition to Eqs. (11) and (12) we also have, along a streamline in the wake, pw p þ 0:5q2w ¼ w1 þ 0:5q2w1 , (20) r r where the w subscript refers to the wake side of the screen and the w1 refers to the downstream wake. Downstream, we can assume that pw1 ¼ p1 . Furthermore, if the fluid is of infinite extent we can assume that U 1 ¼ U 1 ¼ U and consequently that p1 ¼ p1 . Thus we can show that ps pw ¼ 0:5rðv2w u2s þ U 2 q2w1 Þ,
(21)
where by mass conservation, we have assumed that us ¼ uw . Again either Eq. (8) or (9) provides the relevant boundary conditions. They define a function E, which for the case of a screen normal to the flow can be interpreted as a deflection coefficient, and which satisfies vw ¼ Evs and E ¼ us =ð2 us Þ. Square law: On substituting the pressure drop square law into (21) we can show that on a multi-panel screen the following non-dimensional condition must be satisfied on each element, ku2s ¼ ð1 u2s =ð2 us Þ2 Þð1 v2s Þ.
(22)
This is equivalent to the general formulation of Koo and James for the particular case of a finite screen normal to an infinite flow. pffiffiffiffiffi If we consider a single panel model where here we assume that vs ¼ m= 12 we find 3ku4s ð12k þ 4Þu3s þ 12ðk þ 1Þu2s 8 ¼ 0.
(23)
For the case where vs ¼ 0 we have ku4s 4ku3s þ 4ku2s þ 4us 4 ¼ 0,
(24)
which when solved numerically produces the same relationship between cd and k as the approximation given by Koo and James. Darcy’s law: If we assume Darcy’s law, the non-dimensional pressure drop condition at the screen, which must be satisfied at each panel of a multi-panel model, is 2aus ¼ ð1 u2s =ð2 us Þ2 Þð1 v2s Þ.
(25)
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The single panel equivalents are ð3a 2Þu3s þ 6ð1 2aÞu2s þ 12aus 4 ¼ 0 pffiffiffiffiffi when we assume vs ¼ m= 12 and au3s 4au2s þ 2ð2a þ 1Þus 2 ¼ 0
(26)
(27)
when vs is negligible. 4.3. Cumberbatch model Cumberbatch (1982) and later Howells and Waechter (1995) employed the hodograph plane and conformal mappings, together with a source distribution along that part of the boundary that represents the screen, to model the flow outside of the wake. These authors have assumed that the wake pressure is a constant and that on the boundary between wake and the free stream pw p þ 0:5q2w ¼ 1 þ 0:5U 21 . (28) r r Here the w subscript refers only to quantities evaluated on the wake boundary, although pw, which is a free parameter of the model, is assumed constant across the wake immediately behind the screen. Under these circumstances we can show that ps pw ¼ 0:5rðU 2 q2s Þ þ p1 pw .
(29)
Square law: The non-dimensional form of this condition with the square law is u2s ðk þ 1Þ þ v2s ¼ 1 cpw ,
(30)
where cpw ¼ ðpw p1 Þ=0:5rU 2 coefficient in the wake, a free parameter of the model, whose value must be supplied. This condition must be met at each panel of a multi-panel model. pffiffiffiffiffi For the case where we assume that vs ¼ m= 12 this condition becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 9 12cpw þ 6k 9kcpw (31) us ¼ 3k þ 4 and when we set vs ¼ 0 we get rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 cp w us ¼ . kþ1
(32)
Darcy’s law: When we apply Darcy’s law the screen boundary condition is u2s þ 2aus þ v2s ¼ 1 cpw .
(33)
Darcy’s law was chosen by Cumberbatch and Howells and Waechter and by setting this condition and employing the multi-panel model, we derive the source panel model that best approximates their analysis.
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pffiffiffiffiffi As before, by setting vs ¼ m= 12, we can show that
us ¼
1 3a þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 12cpw 6a þ 9a2 4
(34)
and when vs ¼ 0 we have us ¼ a þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ 1 c p w .
(35)
5. Results Table 1 summarises all of these models where those shaded are equivalent (or in the case of Cumberbatch’s approach closest) to the models examined in the literature. MATLAB programs were written to solve the multiple and single panel problems, using the Newton-Raphson method for solving non-linear systems of equations and the polynomial roots function respectively. Fig. 2 plots these results in the form of drag coefficient versus pressure drop coefficient. The assumption that vs is negligible is valid for k values up to about 2.5 and for a values up to about 0.75 (light grey lines). For larger k and a values this assumption leads to an underestimate of the screen drag in the case of Taylor’s approach and an overestimate in both the models of Koo and James, and of Cumberbatch (where we have taken cpw ¼ 0:2). There is very good agreement between the multiple panel predictions (black pffiffiffiffiffi lines) and those of the single panel where vs is assumed equal to m ln 2/p or m= 12 (dark grey lines). The choice of using the mean or the root mean square tangential velocity is informed by examining the drag coefficient in the limit as k (or a) tends to infinity (i.e. as the screen tends towards impermeability). Under these circumstances m ! 2 and us ! 0 and in Taylor’s model cd ! 4vs , in Koo and James cd ! 1 v2s and in Cumberbatch cd ! 1 cpw v2s . Only the Cumberbatch model can approximate experimentally obtained drag coefficient estimates of impermeable plates in the limit as k tends to infinity, which we would expect to have a value of at least 2 (Hoerner, 1965; Parkinson and Jandali, 1970). To do this it is necessary to supply an appropriate estimate of the base pressure. We include the experimental data of Taylor and Davies (1944) in each of the plots of Fig. 2 (where for the Darcy law plots we have equated the experimental k values to 2a). These show that a pressure drop boundary condition based on the square law rather than Darcy’s law is more likely to hold for the screens they tested. The Taylor model seems appropriate for ko1 while the model of Koo and James and that of Cumberbatch are both good fits to the data for k values up to about 4 after which they tend to underestimate the data. Surprisingly the latter two models are even better fits when we assume that vs is negligible. This suggests that, for greater k values, the Koo and James model is no longer valid whereas for the Cumberbatch model we must, at the very least, modify the base pressure value.
u2s þ 2aus þ v2s ¼ 1 cpw
ð3a 2Þu3s þ 6ð1 2aÞu2s þ 12aus 4 ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi us ¼ ð1 3a þ 9 12cpw 6a þ 9a2 Þ=4
These are equivalent (or in the case of Cumberbatch, closest) to the models in the literature.
Cumberbatch
Koo and James 2aus ¼ ð1 u2s =ð2 us Þ2 Þð1 v2s Þ
2 2 2 2 þ 24 u2s 16us þ 4 ¼ 0 4 þ lnp2 2 u4s ln2p2 2 þ 16 u3s þ ð1aln2=4Þp 2
au3s 4au2s þ 2ð2a þ 1Þus 2 ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi us ¼ a þ a2 þ 1 cpw
us ¼ 1 a=2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi us ¼ 1 aus =2 u2s þ v2s
Darcy’s law Taylor
2 2 2 2 u4s ln2p2 2 þ 16 u3s þ lnp2 2 þ 24 u2s 16us þ 4 ¼ 0 us ¼ 1=ð1 þ k=4Þ 4 þ ð1kln2=16Þp 2
Single panel, vs ¼ 0
ku4s 4ku3s þ 4ku2s þ 4us 4 ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi us ¼ ð1 cpw Þ=ðk þ 1Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi us ¼ 1 ku2s =4 u2s þ v2s
pffiffiffiffiffi Single panel, vs ¼ m ln 2=p or m= 12
Koo and James ku2s ¼ ð1 u2s =ð2 us Þ2 Þð1 v2s Þ 3ku4s ð12k þ 4Þu3s þ 12ðk þ 1Þu2s 8 ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cumberbatch u2s ðk þ 1Þ þ v2s ¼ 1 cpw us ¼ ð1 þ 9 12cpw þ 6k 9kcpw Þ=ð3k þ 4Þ
Square law Taylor
Multiple panel
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Table 1 A summary of all the models described in the text and plotted in Fig. 2
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square law
cd
Taylor
Koo and James
Cumberbatch
1.6
1.6
1.6
1.2
1.2
1.2
0.8
0.8
0.8
0.4
0.4
0.4
0 0.1
1
10
100
0 0.1
1
10
100
0 0.1
1
10
100
Darcy’s law
cd
Taylor
Koo and James
Cumberbatch
1.6
1.6
1.6
1.2
1.2
1.2
0.8
0.8
0.8
0.4
0.4
0.4
0 0.1
1
10
100
0 0.1
1
10
100
0 0.1
1
10
100
Fig. 2. Drag coefficient cd versus non-dimensional pressure drop coefficient (k or a, see text) for the 18 models summarised in Table 1. The black lines are the multiple panel predictions, the dark grey lines are pffiffiffiffiffi the single panel predictions when vs is set equal to m ln 2/p or m= 12 and the light grey lines are the predictions when vs is assumed negligible. The data points are the experimental data of Taylor and Davies (1944).
6. Discussion We have shown that when we assume a square law pressure drop condition at the screen boundary the Koo and James model and that of Cumberbatch fit the limited data set available for k values up to about 4. It is important to recognise that this range of k values is likely to cover a wide range of important situations. For instance when Rs, the Reynolds number based on the pore dimensions and the flow speed through the screen, XOð103 Þ, Batchelor (1967) and Hoerner (1952) show that k ¼ ðj=ð1 jÞÞ2 where j is the solidity. In which case the models would appear to be valid for solidities of up to 2=3. The reason the Koo and James and the Cumberbatch models cover a broader k range than the Taylor model is probably because they also model the influence of the wake. The Koo and James model is the more sophisticated in that it models the flow field in the wake whereas Cumberbatch only assumes a constant base pressure. The advantage of the Cumberbatch model, however, is that if the base pressure can be expressed as a function of k, it could be used over the whole range of k, including the case as a screen tends to impermeability. Although the Taylor model does not cover as wide a k range as the other two models it still may be useful (Fredheim and Faltinsen, 2001), and using the above arguments would be valid for solidities of up to 1=2.
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Clearly, there is a need for more experimental data in order to carry out a thorough assessment of models of flow through and around screens. The pressure drop coefficient, k (or a) and the drag coefficient, cd, must be obtained individually in separate experiments. To estimate k the experimental design must be such that all the flow is channelled through the screen whereas to estimate cd the design must be such that the flow goes both through and around the screen and approximates the conditions of a finite screen in an infinite flow field. We must also be careful to distinguish between Rl, the Reynolds number associated with the large-scale screen dimensions and the upstream speed, and Rs. Each of the models described here implicitly assumes that Rl is of Oð103 Þor greater. Otherwise the respective singularity approaches to modelling the screen boundary would not be valid. As pointed out by Howells and Waechter (1995) it is not inconsistent for finite screens in an infinite flow to have low Rs flow through a screen together with high Rl flow around it. Under these circumstances, the viscous effects at the screen are more important, Rs pOð102 Þ and the Darcy law pressure drop boundary condition is likely to be more appropriate. On the other hand the square law pressure drop boundary condition is likely to be more suitable when the flow through the screen is dominated by inertial forces and Rs XOð103 Þ.
Acknowledgements This paper has been carried out with financial support from the Commission of the European Communities, specific RTD programme Quality of Life and Management of Living Resources. It does not necessarily reflect the Commission’s view and in no way anticipates its future policy in this area. References Batchelor, G.K., 1967. An introduction to fluid dynamics. Cambridge University Press, Cambridge. Brundrett, E., 1993. Prediction of pressure drop for incompressible flow through screens. Transactions of the ASME Journal of Fluids Engineering 115, 239–242. Carrothers, P.J.G., Baines, W.D., 1975. Forces on screens inclined to a fluid flow. Transactions of the ASME Journal of Fluids Engineering 97, 116–117. Cumberbatch, E., 1982. Two-dimensional flow past a mesh. Quarterly Journal of Mechanics and Applied Mathematics 35, 335–344. Fredheim, A., Faltinsen, O.M., 2001. A numerical model for the fluid structure interaction of a threedimensional net structure. In: Paschen, M. (Ed.), Contributions on the Theory of Fishing Gears and Related Marine Systems, vol. 2. University of Rostock, Germany. Gradshteyn, I.S., Ryzhik, I.M., 1965. Tables of Integrals, Series and Products. Academic Press, New York and London. Hoerner, S.F., 1952. Aerodynamic properties of screens and fabrics. Textile Research Journal 22, 274–280. Hoerner, S.F., 1965. Fluid-Dynamic Drag. Published by the author. Howells, I.D., Waechter, R.T., 1995. Plane irrotational flow against a porous plate. Quarterly Journal of Mechanics and Applied Mathematics 48, 135–156. Koo, J.-K., James, D.F., 1973. Fluid flow around and through a screen. Journal of Fluid Mechanics 60, 513–538.
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Laws, E.M., Livesey, J.L., 1978. Flow through screens. Annual Review of Fluid Mechanics 10, 247–266. Parkinson, G.V., Jandali, T., 1970. A wake source model for bluff body potential flow. Journal of Fluid Mechanics 40, 577–594. Schubauer, G.B., Spanenburg, W.G., Klebanoff, P.S., 1950. Aerodynamic Characteristics of Damping Screens. NACA, Washington, DC TN 2001, pp. 1–39. Taylor, G.I., 1944. Air Resistance of a Flat Plate of very Porous Material. RM ARC 2236. Also published in ‘The scientific papers of G.I. Taylor’. Cambridge University Press, Cambridge, 1963. Taylor, G.I., Davies, R.M., 1944. The Aerodynamics of Porous Sheets. RM ARC 2237. Also published in ‘The Scientific Papers of G.I. Taylor’, Cambridge University Press, Cambridge, 1963.
Ocean Engineering 33 (2006) 1884–1895 www.elsevier.com/locate/oceaneng
Source models of flow through and around screens and gauzes F.G. O’Neill FRS Marine Laboratory, 375 Victoria Road, Aberdeen, AB11 9DB, Scotland Received 14 February 2005; accepted 25 October 2005 Available online 19 January 2006
Abstract The source panel method is used to investigate the singularity models of flow through and around screens that have been developed by various authors. These studies are extended to complete a framework of models in terms of the level of complexity, the nature of the flow in the wake and the form of pressure drop boundary condition. The models that best represent the limited experimental data that is available are identified and the need for more experimental data, in order to carry out a thorough assessment, is noted. r 2005 Elsevier Ltd. All rights reserved. Keywords: Source panel method; Permeable boundaries; Screens; Gauzes
1. Introduction Flows through and around screens, gauzes and porous membranes occur in a wide range of engineering applications such as fishing gears and plankton samplers, parachutes and sails, filtration and sifting systems and in wind tunnels to control and modify the velocity field. Laws and Livesey (1978) survey the available literature on the topic of flow through screens and categorise the studies according to whether they were concerned with (i) the characterisation of the flow properties of the screens, (ii) the effect of a screen on the time–mean velocity distributions and (iii) the turbulence distribution downstream of screens. In this paper we consider in greater detail the second of these categories and, in E-mail address: [email protected]. 0029-8018/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2005.10.009
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particular, examine the singularity models that have been developed to describe the time–mean flow field through and around finite screens in an infinite flow. The first of these was put forward by Taylor (1944) who modelled the flow outside of the wake by considering a screen to be a uniform distribution of centres of resistance and then used Lagally’s theorem to calculate the source strength of an equivalent uniform distribution of sources. Koo and James (1973) ensure that mass and momentum is conserved across the screen and also model the flow in the wake of the screen. By manipulating the stream function to ensure the conservation of mass and momentum across the screen they define a flow that is irrotational everywhere except in the wake region where it is approximately rotational. Cumberbatch (1982) and later Howells and Waechter (1995) employed the hodograph plane and conformal mappings, together with a source distribution along that part of the boundary that represents the screen, to model the flow outside of the wake. Although these models are conceptually similar, insofar as in each case the screen is represented by a singularity distribution, their predictions differ. In this paper we show that these differences are attributable to the relative complexity of the model and assumptions made concerning (i) the nature of the flow in the wake and (ii) the pressure drop boundary condition. In order to eliminate methodological disparities we use the source panel method to model the screen and thus investigate the different assumptions of these authors.
2. The source panel method The source panel method is a well-established technique (especially in the aeronautics industry) for solving potential flow problems. It is particularly suited for modelling flow through permeable surfaces and is essentially the approach used by Taylor, and Koo and James. Here we define a linear element with a constant source distribution to be our basic element. If we consider a distribution of strength m/2p per unit length on the line, g, between points b and c in the complex plane, the complex potential is Z m c F ðzÞ ¼ lnðz gÞjdgj (1) 2p b and the complex velocity is Z dF m c jdgj ¼ . wðzÞ ¼ dz 2p b z g Substituting g ¼ b þ ðc bÞt where t belongs to [0,1] gives Z mjc bj 1 dt , wðzÞ ¼ 2p 0 z b ðc bÞt mjc bj z c ¼ ln 2pðc bÞ zb and with w ¼ u iv, u ¼ RðwÞ and v ¼ IðwÞ.
(2)
ð3Þ
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More specifically, if we consider the complex velocity of a constant source distribution of strength m/2p on the y-axis between points ib and ib, where here b is a positive real number, we have im z ib ln wðzÞ ¼ . (4) 2p z þ ib Substituting z ¼ iy, where y belongs to [0, b] we can show that the velocity at such a panel element is m im by ln , (5) wðiyÞ ¼ þ 2 2p bþy where the sign of the real part (as shown by Taylor, 1944) is such that the normal velocity along one of these linear elements is m/2 in the direction away from the element. With v ¼ IðwÞ, we can show, by straightforward integration that the average tangential speed at a panel element is Z b m by m ln 2 . (6) ln dy ¼ 2pb 0 bþy p Similarly, we can show after substitution and reference tables (Gradshteyn and Ryzhik, 1965) that the root mean square of the tangential speed is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z 1 Z m 1 b 2 by m ln2 q 2 ln dq, dy ¼ 2 2 2p b 0 bþy 2p 0 ð1 þ q Þ m ¼ pffiffiffiffiffi . ð7Þ 12
3. Boundary conditions The drag on a screen is equal to the pressure drop across the screen. The pressure drop is usually expressed in terms of a non-dimensional coefficient k, Dp ¼ 0:5ru2s k,
(8)
where us is the velocity normal to the screen (Taylor, 1944; Koo and James, 1973) (see Fig. 1). A number of authors have identified expressions relating k to solidity and Reynolds number (Brundrett, 1993; Schubauer et al., 1950; Carrothers and Baines, 1975; Hoerner, 1952). However, we will not go into the form of these relationships here. An alternative description of the pressure drop, that has been used by Howells and Waechter (1995) and Cumberbatch (1982), is one based on Darcy’s law where the pressure drop across the mesh is assumed proportional to the flux through the mesh. This can be expressed as Dp ¼ arUus ,
(9)
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vs
vw
U p-∞
1887
uw
us
ps
U∞ p∞
pw
Fig. 1. The s and w subscripts refer to quantities at the free surface of and in the wake of the screen, respectively, while the N and N subscripts refer to quantities upstream and downstream, respectively.
where U is the upstream speed. As for k, a is non-dimensional and will depend on solidity and Reynolds number. (Note for low solidities us tends to U and both these expressions converge for k ¼ 2a.) Bernoulli’s theorem for the steady motion of an inviscid fluid states that p þ 0:5q2 (10) r is constant along a streamline, where p is the dynamic pressure and q is the speed. If we consider the flow around (and through) a screen of finite extent and look at two streamlines, one which goes around the screen, the other which goes through it, we can show that p1 p þ 0:5U 21 ¼ 1 þ 0:5U 21 (11) r r and p1 p þ 0:5U 21 ¼ s þ 0:5q2s , r r
(12)
where the 1 subscript refers to far upstream, the 1 subscript refers to far downstream outside of the screen wake and the s subscript refers to the free surface of the screen.
4. Models of flow through and around screens Here we examine the three different modelling approaches, as determined by the boundary conditions imposed by Taylor, Koo and James, and Cumberbatch. For each approach we consider the cases where the screen is modelled by (i) multiple panel elements, (ii) a single panel element and vs is approximated either by m ln 2/p or pffiffiffiffiffi m= 12 and (iii) a single panel element and vs is assumed negligible. The first of these is the most general. It is equivalent to the model of Koo and James and is the source panel model that best approximates the analysis of Cumberbatch. The third is equivalent to Taylor’s analysis and to an approximation used by Koo and James. The second is an intermediate model and assumes that the tangential velocity can be
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approximated by the mean (in Taylor’s model) or by the root mean square (in the other two). As we see later this choice is informed by examining the drag coefficient in the limit as the screen tends towards impermeability. Furthermore, in each case we investigate both the square and Darcy’s law form of the pressure drop coefficient. 4.1. Taylor model Taylor models the flow outside of the wake by considering the screen to be a uniform distribution of centres of resistance (i.e. a single panel element) and then uses Lagally’s theorem to calculate the source strength of an equivalent uniform distribution of sources. Lagally’s theorem states that a source of strength m/2p at a point in the flow where the velocity is V produces a resistance rVm. For ffia plane pffiffiffiffiffiffiffiffiffiffiffiffiffiffi screen normal to the flow us ¼ U m=2 and the magnitude of V ¼ u2s þ v2s so that at each panel element we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dp ¼ 2rðU us Þ u2s þ v2s . (13) Square law: Equating this resistance to Eq. (8) and non-dimensionalising, we define the multi-panel extension of Taylor’s analysis by ensuring that at each panel qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ku2s ¼ 4ð1 us Þ u2s þ v2s . (14) A simplification of this is to assume that a screen can be modelled by a single panel element. If vs is approximated by its mean value m ln 2/p, we derive the following quartic equation: 2 2 ð1 k2 =16Þp2 4 2p p 3 þ 16 us þ þ 24 u2s 16us þ 4 ¼ 0, 4þ us ln2 2 ln2 2 ln2 2 (15) which can be solved to express us and consequently cd as functions of k. A further simplification, and one which reproduces the analysis of Taylor, is to assume that vs is negligible to give us ¼ 1=ð1 þ k=4Þ. Darcy’s law: If we use Darcy’s law (ie Eq. (9)) we must satisfy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2aus ¼ 4ð1 us Þ u2s þ v2s
(16)
(17)
on each panel element of a multi-panel screen model. As above, if we assume that the screen can be modelled by a single panel and approximate vs with m ln 2/p, we find that at the screen us must satisfy 2 p2 2p þ 16 u3s 4 þ 2 u4s ln 2 ln2 2 ð1 a2 =4Þp2 þ 24 u2s 16us þ 4 ¼ 0. þ ð18Þ ln2 2
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The Darcy law solution analogous to Taylor’s original one, where vs ¼ 0 is us ¼ 1 a=2.
(19)
4.2. Koo and James model Koo and James (1973) propose a formulation that models the flow through, around and in the wake of a screen placed in a parallel-sided channel. By manipulating the stream function to ensure the conservation of mass and momentum across the screen they define a flow that is irrotational everywhere except in the wake region where it is approximately rotational. Here, we apply their theory to the problem of this paper (i.e. a finite screen normal to an infinite flow). In addition to Eqs. (11) and (12) we also have, along a streamline in the wake, pw p þ 0:5q2w ¼ w1 þ 0:5q2w1 , (20) r r where the w subscript refers to the wake side of the screen and the w1 refers to the downstream wake. Downstream, we can assume that pw1 ¼ p1 . Furthermore, if the fluid is of infinite extent we can assume that U 1 ¼ U 1 ¼ U and consequently that p1 ¼ p1 . Thus we can show that ps pw ¼ 0:5rðv2w u2s þ U 2 q2w1 Þ,
(21)
where by mass conservation, we have assumed that us ¼ uw . Again either Eq. (8) or (9) provides the relevant boundary conditions. They define a function E, which for the case of a screen normal to the flow can be interpreted as a deflection coefficient, and which satisfies vw ¼ Evs and E ¼ us =ð2 us Þ. Square law: On substituting the pressure drop square law into (21) we can show that on a multi-panel screen the following non-dimensional condition must be satisfied on each element, ku2s ¼ ð1 u2s =ð2 us Þ2 Þð1 v2s Þ.
(22)
This is equivalent to the general formulation of Koo and James for the particular case of a finite screen normal to an infinite flow. pffiffiffiffiffi If we consider a single panel model where here we assume that vs ¼ m= 12 we find 3ku4s ð12k þ 4Þu3s þ 12ðk þ 1Þu2s 8 ¼ 0.
(23)
For the case where vs ¼ 0 we have ku4s 4ku3s þ 4ku2s þ 4us 4 ¼ 0,
(24)
which when solved numerically produces the same relationship between cd and k as the approximation given by Koo and James. Darcy’s law: If we assume Darcy’s law, the non-dimensional pressure drop condition at the screen, which must be satisfied at each panel of a multi-panel model, is 2aus ¼ ð1 u2s =ð2 us Þ2 Þð1 v2s Þ.
(25)
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The single panel equivalents are ð3a 2Þu3s þ 6ð1 2aÞu2s þ 12aus 4 ¼ 0 pffiffiffiffiffi when we assume vs ¼ m= 12 and au3s 4au2s þ 2ð2a þ 1Þus 2 ¼ 0
(26)
(27)
when vs is negligible. 4.3. Cumberbatch model Cumberbatch (1982) and later Howells and Waechter (1995) employed the hodograph plane and conformal mappings, together with a source distribution along that part of the boundary that represents the screen, to model the flow outside of the wake. These authors have assumed that the wake pressure is a constant and that on the boundary between wake and the free stream pw p þ 0:5q2w ¼ 1 þ 0:5U 21 . (28) r r Here the w subscript refers only to quantities evaluated on the wake boundary, although pw, which is a free parameter of the model, is assumed constant across the wake immediately behind the screen. Under these circumstances we can show that ps pw ¼ 0:5rðU 2 q2s Þ þ p1 pw .
(29)
Square law: The non-dimensional form of this condition with the square law is u2s ðk þ 1Þ þ v2s ¼ 1 cpw ,
(30)
where cpw ¼ ðpw p1 Þ=0:5rU 2 coefficient in the wake, a free parameter of the model, whose value must be supplied. This condition must be met at each panel of a multi-panel model. pffiffiffiffiffi For the case where we assume that vs ¼ m= 12 this condition becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 9 12cpw þ 6k 9kcpw (31) us ¼ 3k þ 4 and when we set vs ¼ 0 we get rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 cp w us ¼ . kþ1
(32)
Darcy’s law: When we apply Darcy’s law the screen boundary condition is u2s þ 2aus þ v2s ¼ 1 cpw .
(33)
Darcy’s law was chosen by Cumberbatch and Howells and Waechter and by setting this condition and employing the multi-panel model, we derive the source panel model that best approximates their analysis.
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pffiffiffiffiffi As before, by setting vs ¼ m= 12, we can show that
us ¼
1 3a þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 12cpw 6a þ 9a2 4
(34)
and when vs ¼ 0 we have us ¼ a þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ 1 c p w .
(35)
5. Results Table 1 summarises all of these models where those shaded are equivalent (or in the case of Cumberbatch’s approach closest) to the models examined in the literature. MATLAB programs were written to solve the multiple and single panel problems, using the Newton-Raphson method for solving non-linear systems of equations and the polynomial roots function respectively. Fig. 2 plots these results in the form of drag coefficient versus pressure drop coefficient. The assumption that vs is negligible is valid for k values up to about 2.5 and for a values up to about 0.75 (light grey lines). For larger k and a values this assumption leads to an underestimate of the screen drag in the case of Taylor’s approach and an overestimate in both the models of Koo and James, and of Cumberbatch (where we have taken cpw ¼ 0:2). There is very good agreement between the multiple panel predictions (black pffiffiffiffiffi lines) and those of the single panel where vs is assumed equal to m ln 2/p or m= 12 (dark grey lines). The choice of using the mean or the root mean square tangential velocity is informed by examining the drag coefficient in the limit as k (or a) tends to infinity (i.e. as the screen tends towards impermeability). Under these circumstances m ! 2 and us ! 0 and in Taylor’s model cd ! 4vs , in Koo and James cd ! 1 v2s and in Cumberbatch cd ! 1 cpw v2s . Only the Cumberbatch model can approximate experimentally obtained drag coefficient estimates of impermeable plates in the limit as k tends to infinity, which we would expect to have a value of at least 2 (Hoerner, 1965; Parkinson and Jandali, 1970). To do this it is necessary to supply an appropriate estimate of the base pressure. We include the experimental data of Taylor and Davies (1944) in each of the plots of Fig. 2 (where for the Darcy law plots we have equated the experimental k values to 2a). These show that a pressure drop boundary condition based on the square law rather than Darcy’s law is more likely to hold for the screens they tested. The Taylor model seems appropriate for ko1 while the model of Koo and James and that of Cumberbatch are both good fits to the data for k values up to about 4 after which they tend to underestimate the data. Surprisingly the latter two models are even better fits when we assume that vs is negligible. This suggests that, for greater k values, the Koo and James model is no longer valid whereas for the Cumberbatch model we must, at the very least, modify the base pressure value.
u2s þ 2aus þ v2s ¼ 1 cpw
ð3a 2Þu3s þ 6ð1 2aÞu2s þ 12aus 4 ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi us ¼ ð1 3a þ 9 12cpw 6a þ 9a2 Þ=4
These are equivalent (or in the case of Cumberbatch, closest) to the models in the literature.
Cumberbatch
Koo and James 2aus ¼ ð1 u2s =ð2 us Þ2 Þð1 v2s Þ
2 2 2 2 þ 24 u2s 16us þ 4 ¼ 0 4 þ lnp2 2 u4s ln2p2 2 þ 16 u3s þ ð1aln2=4Þp 2
au3s 4au2s þ 2ð2a þ 1Þus 2 ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi us ¼ a þ a2 þ 1 cpw
us ¼ 1 a=2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi us ¼ 1 aus =2 u2s þ v2s
Darcy’s law Taylor
2 2 2 2 u4s ln2p2 2 þ 16 u3s þ lnp2 2 þ 24 u2s 16us þ 4 ¼ 0 us ¼ 1=ð1 þ k=4Þ 4 þ ð1kln2=16Þp 2
Single panel, vs ¼ 0
ku4s 4ku3s þ 4ku2s þ 4us 4 ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi us ¼ ð1 cpw Þ=ðk þ 1Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi us ¼ 1 ku2s =4 u2s þ v2s
pffiffiffiffiffi Single panel, vs ¼ m ln 2=p or m= 12
Koo and James ku2s ¼ ð1 u2s =ð2 us Þ2 Þð1 v2s Þ 3ku4s ð12k þ 4Þu3s þ 12ðk þ 1Þu2s 8 ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cumberbatch u2s ðk þ 1Þ þ v2s ¼ 1 cpw us ¼ ð1 þ 9 12cpw þ 6k 9kcpw Þ=ð3k þ 4Þ
Square law Taylor
Multiple panel
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Table 1 A summary of all the models described in the text and plotted in Fig. 2
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square law
cd
Taylor
Koo and James
Cumberbatch
1.6
1.6
1.6
1.2
1.2
1.2
0.8
0.8
0.8
0.4
0.4
0.4
0 0.1
1
10
100
0 0.1
1
10
100
0 0.1
1
10
100
Darcy’s law
cd
Taylor
Koo and James
Cumberbatch
1.6
1.6
1.6
1.2
1.2
1.2
0.8
0.8
0.8
0.4
0.4
0.4
0 0.1
1
10
100
0 0.1
1
10
100
0 0.1
1
10
100
Fig. 2. Drag coefficient cd versus non-dimensional pressure drop coefficient (k or a, see text) for the 18 models summarised in Table 1. The black lines are the multiple panel predictions, the dark grey lines are pffiffiffiffiffi the single panel predictions when vs is set equal to m ln 2/p or m= 12 and the light grey lines are the predictions when vs is assumed negligible. The data points are the experimental data of Taylor and Davies (1944).
6. Discussion We have shown that when we assume a square law pressure drop condition at the screen boundary the Koo and James model and that of Cumberbatch fit the limited data set available for k values up to about 4. It is important to recognise that this range of k values is likely to cover a wide range of important situations. For instance when Rs, the Reynolds number based on the pore dimensions and the flow speed through the screen, XOð103 Þ, Batchelor (1967) and Hoerner (1952) show that k ¼ ðj=ð1 jÞÞ2 where j is the solidity. In which case the models would appear to be valid for solidities of up to 2=3. The reason the Koo and James and the Cumberbatch models cover a broader k range than the Taylor model is probably because they also model the influence of the wake. The Koo and James model is the more sophisticated in that it models the flow field in the wake whereas Cumberbatch only assumes a constant base pressure. The advantage of the Cumberbatch model, however, is that if the base pressure can be expressed as a function of k, it could be used over the whole range of k, including the case as a screen tends to impermeability. Although the Taylor model does not cover as wide a k range as the other two models it still may be useful (Fredheim and Faltinsen, 2001), and using the above arguments would be valid for solidities of up to 1=2.
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Clearly, there is a need for more experimental data in order to carry out a thorough assessment of models of flow through and around screens. The pressure drop coefficient, k (or a) and the drag coefficient, cd, must be obtained individually in separate experiments. To estimate k the experimental design must be such that all the flow is channelled through the screen whereas to estimate cd the design must be such that the flow goes both through and around the screen and approximates the conditions of a finite screen in an infinite flow field. We must also be careful to distinguish between Rl, the Reynolds number associated with the large-scale screen dimensions and the upstream speed, and Rs. Each of the models described here implicitly assumes that Rl is of Oð103 Þor greater. Otherwise the respective singularity approaches to modelling the screen boundary would not be valid. As pointed out by Howells and Waechter (1995) it is not inconsistent for finite screens in an infinite flow to have low Rs flow through a screen together with high Rl flow around it. Under these circumstances, the viscous effects at the screen are more important, Rs pOð102 Þ and the Darcy law pressure drop boundary condition is likely to be more appropriate. On the other hand the square law pressure drop boundary condition is likely to be more suitable when the flow through the screen is dominated by inertial forces and Rs XOð103 Þ.
Acknowledgements This paper has been carried out with financial support from the Commission of the European Communities, specific RTD programme Quality of Life and Management of Living Resources. It does not necessarily reflect the Commission’s view and in no way anticipates its future policy in this area. References Batchelor, G.K., 1967. An introduction to fluid dynamics. Cambridge University Press, Cambridge. Brundrett, E., 1993. Prediction of pressure drop for incompressible flow through screens. Transactions of the ASME Journal of Fluids Engineering 115, 239–242. Carrothers, P.J.G., Baines, W.D., 1975. Forces on screens inclined to a fluid flow. Transactions of the ASME Journal of Fluids Engineering 97, 116–117. Cumberbatch, E., 1982. Two-dimensional flow past a mesh. Quarterly Journal of Mechanics and Applied Mathematics 35, 335–344. Fredheim, A., Faltinsen, O.M., 2001. A numerical model for the fluid structure interaction of a threedimensional net structure. In: Paschen, M. (Ed.), Contributions on the Theory of Fishing Gears and Related Marine Systems, vol. 2. University of Rostock, Germany. Gradshteyn, I.S., Ryzhik, I.M., 1965. Tables of Integrals, Series and Products. Academic Press, New York and London. Hoerner, S.F., 1952. Aerodynamic properties of screens and fabrics. Textile Research Journal 22, 274–280. Hoerner, S.F., 1965. Fluid-Dynamic Drag. Published by the author. Howells, I.D., Waechter, R.T., 1995. Plane irrotational flow against a porous plate. Quarterly Journal of Mechanics and Applied Mathematics 48, 135–156. Koo, J.-K., James, D.F., 1973. Fluid flow around and through a screen. Journal of Fluid Mechanics 60, 513–538.
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Laws, E.M., Livesey, J.L., 1978. Flow through screens. Annual Review of Fluid Mechanics 10, 247–266. Parkinson, G.V., Jandali, T., 1970. A wake source model for bluff body potential flow. Journal of Fluid Mechanics 40, 577–594. Schubauer, G.B., Spanenburg, W.G., Klebanoff, P.S., 1950. Aerodynamic Characteristics of Damping Screens. NACA, Washington, DC TN 2001, pp. 1–39. Taylor, G.I., 1944. Air Resistance of a Flat Plate of very Porous Material. RM ARC 2236. Also published in ‘The scientific papers of G.I. Taylor’. Cambridge University Press, Cambridge, 1963. Taylor, G.I., Davies, R.M., 1944. The Aerodynamics of Porous Sheets. RM ARC 2237. Also published in ‘The Scientific Papers of G.I. Taylor’, Cambridge University Press, Cambridge, 1963.