A study of the flow in an s shaped duct 3

Chapter RADIAL PRESSURE GRADIENT AND CENTRIFUGAL FORCE IN CURVED AND S-SHAPED DUCT♣ 3.1 Introduction Literature review in Chapter shows that swirl flow in a curved duct and S-duct is governed by a centrifugal force and radial pressure gradient force between the side walls The magnitude of these two forces within the duct is dependent on the curvature of the duct For example, in a curved duct of a sharper bend (or smaller curvature ratio), the pressure difference between the side walls is known to be larger in magnitude than one with a gentler bend Therefore, the side wall pressure variations, illustrated either as plots of CP versus the duct’s turning angle ( ) or the duct’s stream-wise distance (s), generally show significant variations for curved ducts of different curvature ratio Since the dominant forces in a curved duct are due to radial pressure gradient and centrifugal effects, it can be argued that plotting the experimental data as a non-dimensional parameter consisting of a ratio of these two forces should give a better collapse of the data, regardless of the curvature ratio The main objective of this chapter is to investigate this proposition by introducing a dimensionless parameter, , which is related to the ratio of the two stated forces, and apply it to the flow in circular and square cross sectioned 90O curved ducts and S-shaped ducts ♣ A major part of this chapter has been published in Physics of Fluids (2008, Vol.20, Issue Art no 055109) under the title of “On the relation between centrifugal force and radial pressure gradient in flow inside curved and Sshaped ducts” by Ng YT, Luo SC, Lim TT and Ho QW 29 There are a few well known non-dimensional numbers for fluid flow involving curvature effects, e.g Goertler number for curved boundary layer flows and Dean number for curved pipe flow To the best of the author knowledge, it is relatively new to define explicitly the parameter and use it to study curved duct flows of different curvature ratios A better understanding of and its variation in curve ducts may lead to improved design in, say, elbow flow meters (see Hanssen (1980), Borresen (1980) and Sanchez Silva et al (1997)) which uses the magnitude of the radial pressure difference between the side-wall for calibration against the flow speed Later in this chapter, a method to compute is shown It is based on the known side wall CP data from circular and square cross-sectioned 90O curved ducts and S-shaped ducts that are available both in the literature and from the present experimental work 3.2 Experimental Set-up and Methodology The experimental set-up and technique used had been described in Chapter and only a very brief outline will be given here Essentially, the S-duct wind tunnel, and Test Section to were used The flow velocities were Um = m/s and 15 m/s thus giving Reynolds number (Re), based on hydraulic diameter (D = 0.15 m), of 4.74x104 and 1.47x105, respectively The inlet boundary layer thicknesses were 0.05D at the two above mentioned Reynolds numbers The Scanivalve system measured the side wall surface pressure (hence Cp) distribution Pitot-static tube measured the total pressure coefficient (CPT) distribution and crossed hot-wires measured the normalized cross flow velocities (v/Um and w/Um) at the S-duct exit plane Flow visualisation using smoke wires were used to visualize the flow separation phenomenon on the near side wall and vortex generators were used to suppress these flow separation 30 Besides the present experimental data, Cp data on 90O curved ducts and S-ducts of square and circular cross sectioned were extracted from literature Fig 3.1 shows the nomenclature used in this part of the work The far and near side walls are labeled with the origin of the s-axis located on the inlet of the 90O curved duct From the literature, the side wall CP data plots were scanned digitally into Bitmap format and the data points were acquired using a plot digitizer Polynomial curve fits were subsequently applied to these acquired CP data to obtain the intermediate data points Table provides a summary of the test conditions in these referenced cases, detailing their duct geometry, curvature ratio, flow conditions and inlet boundary layer thicknesses (where available) The 90O curved ducts and S-shaped ducts have square and circular constant crosssections, but of different curvature ratios The selected cases are limited to moderately high Reynolds number (Re) (i.e Re ~ O(104) to O(105)) and where the side wall pressures were measured in the plane of the bend In these references, the side wall pressure coefficient data are typically presented as graphs of CP versus (i.e the duct’s turning angle) or CP versus the duct’s non-dimensional centre-line distance Particular attention was paid to the different coordinate system and nomenclature used by different authors in their work For example, all authors define CP in the usual manner as stated above except Ito (1960) who defines a loss coefficient as H , Um 2g where H is the static pressure head loss measured with respect to a reference value Clearly, Ito’s (1960) definition is consistent with those of other authors’ if both the denominator and numerator are multiplied by g 31 In the next section, the Cp data from the present experiment and from the literature will be presented A method to compute the ratio of radial pressure gradient to centrifugal force will be discussed 3.3 Results and Discussion 3.3.1 Downstream Variation of Side Wall Cp The CP data gathered from the literature are presented in Fig 3.2 and Fig 3.3 for 90O curved duct and S-duct respectively For clarity, the present experimental CP data on square cross-sectioned S-ducts are plotted separately in Fig 3.4(a) and (b) for Re = 4.73x104 and Re = 1.47x105 respectively These six figures show the variation of the side wall CP with the normalized axial (or centre line) distance (s/SO) for circular and square cross sectioned 90O curved ducts and S-ducts of different curvature ratio SO is the total centre-line coordinate distance along the s-axis of each curved 90O duct or S-duct For the ease of identification, the CP curves corresponding to near and far side data in Fig 3.2 and 3.3 are circled and labeled From these figures, a few common trends are noted Firstly, for the flow in those 90O curved ducts (in Fig 3.2(a) and 3.2(b)), the far side wall has relatively higher CP values than the near side wall, and this is also true for the CP distribution in the first bend of those S-shaped ducts (Fig 3.3(a) and 3.3(b), 3.4(a) and 3.4(b)) Secondly, for the flow in S-shaped ducts, the CP distribution exhibits a sinusoidal-like variation along the ducts’ centre-line distance This implies that the pressure difference between the side walls changes sign as the flow negotiates the bends of opposite sense This well known flow behavior results in a swirl development in the first bend, which is subsequently attenuated and reversed (in direction) in the second bend This observation is generally true for flows in circular and square cross-sectioned S-duct and is reflected in the CP 32 distribution displayed in Fig 3.3(a) (for circular cross-sectioned S-duct) and in Fig 3.3(b), 3.4(a) and 3.4(b) (for square cross-sectioned S-duct) Thirdly, these figures also show that with increasing curvature ratio (RC/D), the ducts’ geometry becomes less curved and the radial pressure difference, p (defined as the pressure difference between the far side and near side wall at the same s-coordinate) decreases This is clearly observed in Fig 3.2(a) and (b) For example, in Fig 3.2(a), the CP difference between the side walls in Briley et al.’s (1982) data at RC/D = 1.0 is larger than those at RC/D = 2.8 The same can be said of Ward-Smith’s (1971) data in Fig 3.2(b), when one compares the RC/D = 1.15 data with those at RC/D = 3.45 This is due to centrifugal force of different magnitudes exerting on the fluid as it flows around a bend A sharper bend (and hence lower value of Rc/D) gives rise to higher centrifugal force and hence, as a reaction force, a larger pressure difference develops between the side walls Lastly, in Fig 3.4(a) and (b), the experimental CP distribution for the flow in square cross sectioned S-duct shows a point of inflection which indicates that flow separation is present Evidence of flow separation will be given in a later section From the above discussion and the figures presented, it is clear that the variation of side wall CP with s/SO is dependent on duct curvature ratio and cross sectional shape, and is likely to result in the large scatter among the published CP data Since the forces due to radial pressure gradients and centrifugal effects are the dominant forces governing the physics of the flow in a bend, it appears likely that a parameter involving the two above-mentioned forces may be able to put ducts of different curvature on a “common ground” If one goes back to the basic physics of the fluid flow, for the present problem, it is reasonable to say that a force, F, experienced by the fluid is dependent on the duct dimension D, inlet mean velocity Um ,fluid density , dynamic viscosity , duct radius of curvature Rc and pressure difference between inner and outer wall p 33 That is, F = f(D, Um, , , Rc, p) Using Buckingham Theorem, it can be shown that the above parameters can be reduced to four dimensionless groups, namely Π1 = Π2 = F , ρU m D µ ρU m D = , Re Rc , and D ∆p Π4 = ρU m Π3 = Here, a term that involves both the radius of curvature of the bend Rc and the pressure difference between the inner and outer walls would be a combination of this new dimensionless parameter is termed and In this thesis, and defined as ∆p Ω = Π3 × Π = ρU m D (3.1) Rc Note that the numerator is related to pressure gradient force between the side walls of the curved duct while the denominator is related to centrifugal force In terms of a better known dimensionless group called the Dean’ s number, which is defined as De = D Re , it can be 2Rc shown that, Ω= ∆p Re 2 ρU m De (3.2) A simple method of calculating Ω from the curved ducts’ known side wall CP values and its curvature ratio is illustrated in the next section This derived relation was subsequently applied to all data in the present work 34 p 3.3.2 Determination of = Um D Rc To compute Ω from the available CP data, it is first assumed that the reference static pressure (p ) is constant within each of the cases obtained from literature This is a reasonable assumption because a constant reference pressure (usually the wall static pressure) is used in experiments Next, the difference in CP between the far side wall and near side wall can be written as, Cp far − Cp near = By multiplying a factor of (C p far − Cp p far − p∞ p − p∞ − near , 1 2 ρU m ρU m 2 = ∆p ρU m (3.3) Rc to both sides of the equation (3.3), the relation becomes, 2D ) 2RD = ∆p c near ∆p = ρU m ρU m D = Rc 2D (3.4) Rc Hence, the value of Ω at a particular (s/SO), is equal to the difference in CP between the far side and near side wall at the same (s/SO), multiplied by half of the curvature ratio This mathematical operation is applied to all the CP data and presented as plots of (s/SO) versus (s/SO) The clear advantage of illustrating the plots in such a manner is that it allows one to study the variation of these two forces along the normalized curved ducts’ centerline coordinate (or 35 distance) for ducts of different curvature ratios In addition, the scattered data of CP versus (s/SO) for curved ducts of different curvature ratio are reduced to collapsed curves when plotted in the proposed manner In the following sections, salient points in the collapsed curves of (s/SO) versus (s/SO) for the flow in square and circular cross-sectioned 90O curved ducts and S-ducts are discussed 3.3.3 Collapsed Curves of Ω (s/SO) Versus (s/SO) 3.3.3.1 Circular and Square Cross-sectioned 90OCurved Ducts Fig 3.5(a) and (b) show the variation of (s/SO) with (s/SO) for flows in circular and square cross-sectioned 90O curved ducts Included in each of these figures is a straight line of unit gradient Data points which lie close to this line imply that the force due to radial pressure gradient is balanced by that due to centrifugal effects In Fig 3.5(a) and (b), the various literature data for circular and square cross-sectioned 90O curved duct show that respective collapsed curves are obtained for ducts of different curvature ratios The solid line in each figure depicts the averaged variation of (s/SO) with (s/SO) for the two flow cases considered The figures also show that the variation of averaged (s/SO) with (s/SO) remains fairly linear up to (s/SO) 0.6, suggesting that the radial pressure gradient force is balanced by the centrifugal force as the flow negotiates the bend Beyond that, the curve attains a maximum point at (s/SO) 0.7-0.8 and decreases thereafter Increased data scatter are noted in the two figures when (s/SO) > 0.6 This could be due to increased flow unsteadiness and the presence of flow separation as the flow approaches the exit of the bend From the points noted above, the similarity in the variation of the averaged (s/SO) with (s/SO) for flows in square and circular cross-sectioned 90O curved ducts at different curvature ratios signals the importance of the radial pressure gradient and 36 centrifugal force in influencing the flow characteristics Comparing Fig 3.5(a) and (b) with Fig 3.2(a) and (b), it is clearly advantageous to plot (s/SO) with (s/SO) instead of side wall CP with (s/SO) since the scatter in the CP data as depicted in the latter figures is significantly reduced (and in the earlier part of the flow (s/SO