Turbulence has been studied by scientists and engineers for decades as it appears in the majority of the fluids existent in nature and in engineering applications and because turbulent flow and its underlying behaviour are tremendously complex. The University of Manchester is widely viewed as the birthplace of turbulence due to the pioneering work of one of its prominent academics, Professor Osborne Reynolds (1842 -1912). Building on this legacy, a classical experimental apparatus has bee n used in this paper to study a turbulent pipe flow with the aim of measuring the mean velocity field and wall shear stress using four experimental techniques, all developed in the 20th century, namely static pressure drop; mean square signals measured fro m a hot -wire; Preston tube; and the ‘Clauser Plot’. The experimental results have then been compared against those obtained using Computational Fluid Dynamics (CFD), utilising different two -equation turbulence models. The present work highlights the discre pancies evident in obtaining the value of the wall shear stress in each method. In addition, the scopes and limitations of each technique are discussed in detail, highlighting the clear evolution of turbulence study tools over the last 100 years. Keywords : CFD - experimental fluid mechanics - fully -developed pipe flow - turbulence - Preston tube - Clauser plot Nomenclature 𝐸𝑎𝑐̅̅̅̅ mean voltage alternating current [V] 𝐸̅ mean voltage across the hot -wire [V] 𝑢′̅ mean velocity fluctuation [m/s] 𝑈̅ mean v elocity [m/s] Ac cross sectional area of the pipe [m2] Cf wall skin friction coefficient 𝐷 outer diameter of Preston tube [m] 𝑑 inner diameter of Preston tube [m] 𝐸 voltage across the hot -wire [V] 𝐸1 output voltage of the 45o hot-wire anemometry in positive traverse [V] 𝐸2 output voltage of the 45o hot-wire anemometry in negative traverse [V] 𝐸𝑟𝑚𝑠 root mean square voltage [V] 𝑘 turbulent kinetic energy [m2/s2] 𝐿 distance between static tapping points [m] 𝑙 characteristic length [m] 𝑙𝑚 mixing length [m] 𝑛 power -law exponent 𝑅 pipe radius [m] 𝑟 distance from the centre [m] 𝑅𝑒 Reynolds number 𝑅𝑒𝑐 critical Reynolds number 𝑈 total velocity [m/s] 𝑈+ dimensionless velocity 𝑈𝜏 shear velocity [m/s] 𝑈𝑚 maximum velocity at the centre line [m/s] 𝑈𝑟𝑚𝑠 root mean square velocity [m/s] * Gabriela Lopez -Santana - Andrew Kennaugh - Amir Keshmiri ( ) Department of Mechanical, Aerospace and Civil Engineering (MACE), The University of Manchester Manchester, M13 9PL, United Kingdom Tel.: +44(0)161 306 5752 email: a.keshmiri@manchester.ac.uk 𝑌+ dimensionless near wall distance 𝛥𝑃 pressure differences [Pa] 𝜇 coefficient of dynamic viscosity [Ns/m2] 𝜈 coefficient of kinematic viscosity [m2/s] 𝜔 specific rate of dissipation [l/s] 𝜌 density [kg/m3] 𝜏 total shear stress [Pa] 𝜏𝑥𝑦′ Reynold shear stress measured by a hot -wire method [Pa] 𝜏𝑤𝑃𝐷 wall shear stress measured by the static pressure drop metho d [Pa] 𝜏𝑤 wall shear stress [Pa] 𝜀 turbulence dissipation rate [m2/ s3] 𝑘ℎ𝑤 hot-wire calibration coefficient [V/m/s] 1 Introduction 1.1 Background Turbulent flow is amongst the most studied subjects in fluid dynamics, largely due to its presence across a wide variety of applications. A flow is characterised as turbulent when the Reynolds number exceeds a critical value. The critical Reynolds number, as well as the characteristic length, are case -specific parameters. Common environ mental and industrial applications are ass ociated with augmented Reynolds numbers which characterise the ﬂow regime as turbulent [1]. The present investigation aims to study the behaviour of the ﬂuid inside a pipe: air was forced to ﬂow inside by the action of an upstream fan for the quantiﬁcation of the mean velocity ﬁeld and the estimation of shear stresses of fully -developed turbulent ﬂow. An additional purpose of this work is to highlight the evolution of turbulent ﬂow analysis tools by applying common experimental and computational techniques. Speciﬁcally, hot -wire anemometry [2] was used to measure the mean and ﬂuctuating properties of a turbulent ﬂow, while the wall shear stress was measured using four different methods: a) Static pressure drop over a known length of pipe [3]; b) Mean square signals from a hot -wire [4]; c) Preston Tube [5,6]; and, d) The law of the wall (Clauser Plot) [7,8]. The velocity and viscosity measurements were further analysed with attention given to the distribution across the pipe. The commercial CFD software, ANSYS Fluent[9], was used to numerically simulate the turbulent ﬂuid ﬂow in the pipe. The results of all the methods used here are discussed in detail, outlining from historical and practical perspectives the evolution of techniques available for a ﬂuid dynamic ist studying the complex phenomena involved in turbulent ﬂows. 1.2 Historical Evolution of the Experimental and Computational Techniques Experimental techniques have been used to study turbulent ﬂows for many decades. The experimental work of Professor Reynolds [10] in the late 19th century at the University of Manchester contextualised the parameters associated with turbulent ﬂows. A substantial number of studies have speciﬁcally examined turbulent pipe ﬂows and it is of particular interest to note the evolution of these methods over time. The static pressure drop method, based on the fundamental theory of ﬂuid mechanics, has been applied for the wall shear stress calculation since the introduction of pressure tapping techniques in 1888 [17]. H ot-wire anemometry was introduced at the beginning of the 20th century based on the theoretical work of Boussinesq in 1905 [18] and Russel in 1911 [19], although the ﬁrst actual experiments were not conducted until the 1910s, most notably in the signiﬁcant work of King in 1914 [20]. Subsequently, research on the measurement of ﬂuctuating turbulent velocity with hot -wires in 1929 [21] enabled the application of hot -wire anemometry for the calculation of the wall shear stresses. In 1954, Preston [6] developed a new approach on the principle that the surface shear stress can be accurately predicted by modiﬁed Pitot tubes of different sizes. Finally, the law of the wall, also known as the Clauser Plot, was introduced in 1956 [7] offering a graphic appr oach for approximating the wall shear stress. Since 1969, computational techniques have discretised the Navier -Stokes equations in space and time in order to investigate turbulent effects [11]. However, due to the very ﬁne structures associated with turbul ent ﬂows, the analytical solution offered by the Navier -Stokes equations is hardly feasible [12]; nevertheless, in many cases, the ﬁne turbulent structures may be of low interest. As such, approximation models have been developed for the estimation of turb ulent ﬂow ﬁelds [13 –16]. The ﬁrst computational technique examining turbulence was the Large Eddy Simulation (LES), introduced in 1970 by Deardorff [16]. In 1972, Orszag and Patterson developed the ﬁrst Direct Numerical Simulation (DNS) for low Reynolds nu mber [22]. Despite their high level of detail, they are limited by their requirements of supercomputers and by their computational times, which are excessively long for everyday applications [23]. Launder and Spalding presented the Reynolds -averaged Navier -Stokes (RANS) in 1972, which went on to become the most common technique to model turbulence, even until today. Its improved efﬁciency has made the RANS method the standard tool for turbulence modelling for most engineering applications [24 –30]. 1.3 Aims and Objectives The main aim of the present work is to assess the accuracy of four classical experimental techniques, all developed in the 20th century for measuring the mean velocity ﬁeld and wall shear stress for a fully -developed pipe ﬂow. The experimental data are then compared against the CFD results utilising a two -equation eddy -viscosity turbulence model to provide an assessment on the performance of historic experimental techniques with one of the latest computational simulations. 2 Methods 2.1 Test Rig and Instrumentation The experimental set -up used in the present study is located in the premises of The University of Manchester, shown in Fig. 1. It consists of : (a) Equipment at the pipe exit. (b) Digital devices used in this experiment. Fig. 1: Test rig for the study of fully -developed turbulent ﬂow in a pipe. • A brass pipe with length to diameter ratio of 125:1. This length to diameter ratio is considered adequate for a high Reynolds number. • A variable speed centrifugal fa n that is used at the inlet and forces air at ambient conditions to flow along the pipe. • An analogue manometer which is connected via a pressure switch to static pressure tapings in the wall of the pipe and also to a Preston tube placed at the pipe exit. • A Preston tube that is used at the outlet of the brass pipe in order to measure the dynami c pressure. • Static pressure tappings that are used at two points on the wall of the pipe to calculate the pressure drop along the pipe over a ﬁxed distance. • A set of two pressure switches that are used to interconnect a digital manometer apparatus with the Preston tube. • A hot -wire anemometer. 2.2 Instrumentation and Measurements 2.2.1 Anemometry measuring system • Two single hot -wire probes (one straight and one 450 hot-wire). • A set of traverse gears are implemented to move the hot -wire to the required position in the pipe. • Two Constant Temperature Anemometer (CTA) bridge s. • A linearizer is used to enhance the non -linear behaviour of the ampliﬁer and to ﬁlter down a portion of the noise interfering with the output data from hot -wire anemometry wh ich is mainly caused by the ﬂuctuation of the probe in the pipe. • The mean and the Root Mean Square (RMS) velocities are measured by two separate voltage meters. • Two digital voltmeters. • AC ampliﬁer for increasing the signal of the time -varying voltage. 2.2.2 Hot -Wire Anemometer Hot-wire anemometers have been widely used to measure velocities, especially those with high frequency velocity ﬂuctuations, i.e. turbulence [31]. A hot -wire anemometer consists of a sensor used to measure the change of velocities r apidly with good spatial and time resolution and comprises a metallic (usually tungsten) ﬁlament of a micron in diameter [32] that has a known and constant variation of resistance with temperature. This ﬁlament is welded to metallic prongs of much larger d iameter and the ﬁlament length is typically of the order of 2 - 3mm long depending on the probe design. When used, the hot -wire probe is placed in air streams and is heated by passing a small electric current through it and some of the heat generated by Jo ule effect in the ﬁlament is absorbed by the air ﬂow via convection [33]. This cooling effect changes the resistance of the wire and thus provides a means of measuring the ﬂow since the change of resistance reﬂects the ﬂow velocity. There are two main types of hot-wire anemometer, namely the constant current (CCA) and constant temperature (CTA); the latter is more common. In operation, the CCA is continuously cooled by the ﬂow and this ultimately leads to the sensitivity decreasing as the wire temperatu re approaches ambient at high velocities. With the CTA, there is a servo electrical system consisting of a Wheatstone bridge that responds to the change of resistance of the hot -wire by either increasing or decreasing the power to it according to whether it is too cold (high velocity) or too hot (low velocity) [34]. This change of power is measured as a change of voltage that can be directly related to the ﬂow velocity by suitable calibration [33]. It is essential that the hot -wire does not make contact wit h the pipe wall, as this renders it ineffectual; it was observed that the hot -wire measurements near to the wall increase the zero volts and any subsequent measurement produces an error [35] and also that the hot -wire is shorted out if both prongs touch the wall. The hot -wire probes here were used with an Analogue to Digital Converter (ADC) and a computer control, which allowed ﬂuctuations in time to be recorded from the sensor. 2.3 Experimental Procedure 2.3.1 Measurement of the Velocity Field The hot-wire was positioned so that measurements of the mean and RMS velocities could be appropriately acquired from a distance as close to the wall as possible using the digital voltmeters. The only way to ensure positional accuracy is to connect a voltmeter to measure the resistance between the hot -wire and the brass tube. The contact between them means that it is at the wall (i.e. 0 mm). Preliminary measurements were taken from an initial distance of 0 .1mm from the wall and, subsequently, several measurements were taken along the radius of the tube up to the centre -line, enabling the average Reynolds number, which determines the regime of the ﬂow in the pipe, to be calculated. Usually, without using the linearizer, the calibration follows King’s law [36]. 𝐸2= 𝐴+𝐵𝑢𝑛 (1) Where E is the voltage along the hot -wire, u is the velocity of the normal ﬂow of the wire and A, B and n are constants. In this experiment, the straight hot -wire was pre -calibrated with a linear calibration coefﬁcient 𝑘ℎ𝑤 equal to 0.5 𝑉/𝑚𝑠−1 applied to the hot -wire output. The calibration coefﬁcient is non -linear, however here a linear slope is assumed, meaning that, to calculate the velocity components, the readings from the voltmeter need t o be multiplied by a factor of two. The required velocities were obtained as follows: 𝑈̅= 𝐸𝑎𝑐̅̅̅̅1 𝑘ℎ𝑤 (2) 𝑈𝑟𝑚𝑠 = 𝐸𝑟𝑚𝑠1 𝑘ℎ𝑤 (3) 𝑈=𝑈 ̅+ 𝑈𝑟𝑚𝑠 (4) Equation (4) adopts the Reynolds decomposition, which represents the instantaneous velocity as a summation of the mean (time -averaged) velocity and the ﬂuctuating velocity. The convention of associating the x -axis with the mean velocity and y -axis with the direction traverse to the supports is used to represent the velocity. It is worth noting that the methods introduced above are only valid when the ﬂow in the pipe is fully -developed and turbulent. Thus, to verify this condi tion, it is necessary to initially calculate the Reynolds number using the non -dimensional Reynolds equation: 𝑅𝑒=𝜌𝑈𝐷𝑝𝑖𝑝𝑒 𝜇 (5) The distribution of the velocity proﬁle gives a reliable illustration of the main features of the turbulent ﬂow and can be compared with the data collected in experiments utilising dimensionless analysis. Dimensionless velocities indicate that there is a general solution of the velocity proﬁle applicable for all ﬂuid and is calculated a s follows: 𝑈+=𝑈 𝑈𝜏 (6) where the shear velocity, 𝑈𝜏, is deﬁned as: 𝑈𝜏=√𝜏𝑤 𝜌 (7) where 𝜏𝑤 is the wall shear stress and 𝜌 is the density . The dimensionless near wall distance, 𝑌+, is deﬁned as: 𝑌+=𝑦𝑈𝜏 𝜈 (8) where 𝑦 is the distance to the nearest wall and 𝜈 is the kinematic viscosity. 2.3.2 Power -law velocity proﬁle The power -law velocity proﬁle is used to represent turbulent ﬂow in a pipe. The empirical relation is obtained as: 𝑈̅ 𝑈𝑚=(1−r R)1 n (9) where 𝑈̅ is the mean velocity, 𝑈𝑚 is the maximum velocity at the centre line, 𝑟 is the distance from the pipe centre, 𝑅 is the total radius of the pipe in this case 𝑅=51𝑚𝑚 and n is a constant exponent that depends on the Reynolds number. The empirical one-seventh power -law declares that the value of 𝑛=7 approximates most of the ﬂows [37]. Method 1: Static Pressure Drop The static pressure drop method, available for fully developed internal ﬂows and axisymmetric conditions, enables the measurement of the average wall shear stress from the axial pressure gradient. The pressure drop is measured with the use of two static pressure tapings along the pipe. The shear stress 𝜏𝑤𝑃𝐷 for a fully developed circular pipe acting over a surface area 𝐴 is related with the pressure drop Δ𝑃 over a length 𝐿 of a constant cross -sectional area 𝐴𝑐 by: ∫𝜏𝑤𝑃𝐷 𝑑𝐴= ∆𝑃𝐴𝑐 𝐴 (10) 𝜏𝑤𝑃𝐷 =∆𝑃𝜋𝑅2 2𝜋𝑅𝐿=∆𝑃 2𝐿𝑅 (11) Harotinidis [38] suggested that this method is perhaps the most reliable for ﬂows satisfying the conditions listed above. In fact, only ﬂows in a circular pipe would meet the above conditions; and, even for a fully developed pipe ﬂow, this method affects the accuracy of the measurement, due to the size of the pressure tap and the quality of the tap hole [39]. Method 2: Mean Square Signal (45° Hot -wire) The slanted hot -wire is sensitive to the components of the velocity ﬂuctuation of turbulent and/or unsteady ﬂow. The wire orientation with respect to the mean ﬂow vector and wire cooling laws vary the sensitivity coefﬁcients, A. The equation that describes the heat transfer of the hot -wire can be linearized to establish a relation between the anemometer voltage ﬂuctuat ion, E, and the velocity ﬂuctuations, 𝑢′,𝑣′,𝑤′, if these ﬂuctuations are minor [39]. 𝐸= 𝐴𝑢𝑢′+𝐴𝑣𝑣′+ 𝐴𝑤𝑤′ (12) After squaring and averaging the equation, the mean value of the anemometer ﬂuctuations can be found: 𝐸2̅̅̅̅= 𝐴𝑢2𝑢′2̅̅̅̅+𝐴𝑣2𝑣′2̅̅̅̅+𝐴𝑤2𝑤′2̅̅̅̅̅+2𝐴𝑢𝐴𝑣𝑢′𝑣′̅̅̅̅̅+2𝐴𝑢𝐴𝑤𝑢′𝑤′̅̅̅̅̅̅+2𝐴𝑣𝐴𝑤𝑣′𝑤′̅̅̅̅̅̅ (13) Deriving the sensitive coefﬁcients of a slanted hot -wire and assuming steady ﬂow and simple cooling laws, the heat transfer of a hot-wire by an effective velocity can be described by King’s law, where the slope represents the sensitivity [36]. In this method, the wall shear stress was estimated using a single 450 hot-wire in reversed horizontal attitudes, positioned half a diameter upstream of the pipe exit. The initial probe position was set to 10 mm from the wall and then traversed the diameter of the pipe towards the centre line, using pre -deﬁned increments. A second set of measurements was taken at the same positions with the probe rotated 1800. On this occasion, the probe traversed the diameter of the pipe from the centreline towards the wall until it reached the distance of 10 mm from the wall. The RMS voltages were measured with the 450 hot-wire, which translate the RMS velocities. The two traverses in the horizontal plane, 𝐸1 and 𝐸2, correspond to the instantaneous voltage applied across the hot - wire [34]. 𝑢′𝑣′̅̅̅̅̅= 𝐸22−𝐸12 2𝑘ℎ𝑤2 (14) Where 𝑢′𝑣′̅̅̅̅̅ represents the time average of the product of the ﬂuctuating velocity components 𝑢’ and 𝑣’ and has the dimension of velocity squared; 𝑘ℎ𝑤 is the calibration coefﬁcient mentioned earlier. Due to the eddy motion of the particles of the ﬂuid, the Reynolds shear stress can be interpreted as the shear forc e per unit area and, to calculate the wall shear stress, the following equation can be used: 𝜏𝑥𝑦′= 𝜌 𝑢′𝑣′̅̅̅̅̅ (15) the boundary conditions applied at the wall are : (𝑢′𝑣′̅̅̅̅̅̅)𝑤𝑎𝑙𝑙 =0 and 𝑘𝑤𝑎𝑙𝑙 =0. Based on McDonough’s work [41] demonstrating that the shear stress is a linear function of the variable radius r and the wall shear stress, the value of the shear stress can also be obtained through: 𝜏= 𝜏𝑤𝑟 𝑅 (16) Method 3: Preston Tube The wall shear stress was also measured with the use of a Preston tube, positioned so that its outer surface was tangential to the pipe inner wall while facing upstream. With appropriate conﬁguration of the analogue manometer, the dynamic pressure can be taken. Theoretically, with the Preston tube, the wall shear stress is related to the dynamic pressure as: [𝜌𝜏𝑤𝑑2 𝜇2]=𝑓[𝜌∆𝑃𝑑2 𝜇2,𝑑 𝐷] (17) where d and D are the internal and external diameters of the Preston tube respectively. As the Preston tube is a thick -walled tube, it tends to give the most reproducible behaviour because it matches the skin friction with the data of the stagnation pressure from the tube placed at the wall. Historical tests have established relationships between the wall shear stress 𝜏𝑤 and the dynamic pressure Δ𝑃, deﬁned as the mean difference between the pressure sensed by the Preston tube and the static pressure obtained from a nearb y wall tap [39]. In this case, where 𝑑 𝐷=0.6, the following widely accepted analytical relationships were obtained [42]: 0<𝑋∗<2.9 𝑌∗=𝑋∗+0.037 (18) 2.9<𝑋∗<5.6 𝑌∗=0.827 −0.1381 𝑋∗+0.1437 𝑋∗2−0.006 𝑋∗3 (149) 5.6<𝑋∗<7.6 𝑌∗=2 𝑙𝑜𝑔 10(1.95𝑋∗+4.10)+𝑋∗ (20) where 𝑋∗=𝑙𝑜𝑔 10[𝜌∆𝑃𝑑2 4𝜇2] (151) Having prescribed 𝑋∗ and determined 𝑌+, the wall shear stress can then be defined as: 𝜏𝑤=4 (10𝑌∗)𝜇2 𝜌𝑑2 (162) Method 4: Clauser Plot In turbulent ﬂows over smooth surfaces, there is always a sub -layer next to the wall in which the ﬂow is laminar, since a sol id surface tends to damp out the turbulent ﬂuctuations. Nevertheless, within the viscous sub -layer and the inner region (which is a fully turbulent ﬂow region), the turbulent stress is zero. In the log -law region at the inner region, the dimensionless velo city is represented in the logarithmic form (see Fig.2b) as: 𝑈+=1 𝜅ln𝑌++𝐵 (17) where 𝜅 is the Von -Karman constant and is assumed 𝜅=0.41, and 𝐵=5.3 [7,8]. This form of velocity distribution is widely used in turbulent pipe ﬂow and is known as the “Universal Law of the Wall” [41]. The Clauser plot is a graphical method used to estimate the turbulent wall shear stress from its correlation with the law of the wall. Equation (23) can be rearranged and presented in a normalised form to obtain the wall shear stress, using the properties of the velocity proﬁle in the logarithmic part of the boundary layer [43]: 𝑈(𝑟) 𝑈𝜏=1 𝜅ln(𝑦𝑈𝜏 𝜈)+𝐵 (184) Multiplying both sides by 𝑈𝜏 𝑈𝑚 gives: 𝑈(𝑟) 𝑈𝑚=[1 𝜅𝑈𝜏 𝑈𝑚] ln(𝑦𝑈𝑚 𝜈)+[1 𝜅𝑈𝜏 𝑈𝑚ln(𝑈𝜏 𝑈𝑚)+𝐵𝑈𝜏 𝑈𝑚] (195) where, as indicated previously, the friction velocity is equal to: 𝑈𝜏=√𝜏𝑤 𝜌 and noting that the friction coefﬁcient is given by: 𝐶𝑓=2(𝑈𝜏 𝑈𝑚 )2 (26) The Clauser plot equation can be rewritten as follows: 𝑈(𝑟) 𝑈𝑚=[1 𝜅√𝐶𝑓 2] ln(𝑦𝑈𝑚 𝜈)+[1 𝜅√𝐶𝑓 2ln(√𝐶𝑓 2)+𝐵√𝐶𝑓 2] (27) 𝜅 and 𝐵 are constants and the velocity terms can be obtained from the experiment, leaving only the friction coefﬁcient as an unknown. The value of the friction coefﬁcient 𝐶𝑓 is estimated graphically when different values are sup erposed in the semi -logarithmic plot of 𝑈 𝑈𝑚 vs 𝑈𝑚𝑦 𝜈 [44] and will be further discussed in Section 3.1.2. Therefore, the wall shear stress can be obtained by rearranging equation (26) and replacing the friction velocity from equation (7): 𝜏𝑤=𝜌𝑈𝑚2𝐶𝑓 2 (208) 2.4 Computational Method In the present work, the fully -developed turbulent pipe ﬂow was simulated computationally using the most popular commercial CFD package, ANSYS Fluent. The length of the pipe in the expe riment was required to be as long as 𝐿=125 𝐷 to ensure a fully -developed turbulent ﬂow. The present computational domain consists of a cylindrical segment of length 4 mm and diameter 102 mm, simulated using a periodic boundary condition. This signiﬁcantly reduces the computational time, and also takes advantage of the axisymmetric pipe. At the entrance of the pipe, an estimated mass ﬂow rate of 𝑚̇=7.5𝑥10−2𝑘𝑔/𝑠 is translated into a velocity of 7.5𝑚/𝑠, which is equa lly distributed along and normal to the inlet surface. In this experiment the Reynolds number was notably high 𝑅𝑒=57469 , so the assumption of a turbulent regime is valid. A turbulence intensity of 10% is assumed at the inlet. No slip -boundary conditions were applied on the pipe wall. A static pressure boundary condition of zero pascals was set at the outlet. 2.4.1 Grid Generation A multi -block grid was generated from the 3D circular pipe computational domain and was divided in two sub - regions: a central square region and an exterior circular region. Accurate representation of the viscous sublayer is required for a satisfactory prediction of wall -bounded turbulent ﬂuids. The ﬁrst cell height was calculated using equation (8). The mesh must be ﬁnest around the wall and become progressively coarser towards the central square region, with an aspect ratio of 1.2. The stream -wise direction does not require an excessive number of cells. In order to analyse grid independency, four different grid resolution s were compared as shown in Table 1. Table 1: Comparative table of 3D Multiblock/Hexahedral grid method. Coarse mesh Medium mesh Fine mesh Very fine mesh 3060 cells 9520 cells 30240 cells 90045 cells 2.4.2 Turbulence Modelling Turbulence modelling is a crucial methodological aspect of any CFD simulation and it signiﬁcantly affects the ﬁnal results. In theory, the Navier -Stokes equations are adequate for simulating a turbulent ﬂow, but, due to the small size and ﬁne detail of the turbulent structures and the consequent demand for a very ﬁne domain discretization, the Direct Numerical Simulation (DNS) is deemed computationally unaffordable. To overcome these profound limitations, turbulence models have been developed in which the e ddies are approximated either by an averaged ﬂow ﬁeld (RANS approach) or by limiting the size of the eddies which are modelled (Large Eddy Simulation (LES)). The turbulence model to approximate the eddies by an averaged ﬂow ﬁeld used in this work is the RA NS approach, since it is the most computationally inexpensive and offers a good representation of the ﬂow ﬁeld [24]. Several RANS models have been introduced in the last decades and applied in demanding CFD applications (e.g. [46 –50]). In this study, the 𝜅−𝜔−𝑆𝑆𝑇 (Shear Stress Transport) turbulence model is used, which is a combination of the 𝜅−𝜔 and the 𝜅−𝜀 turbulence models as proposed by Menter [51]. In this model, the 𝜅−𝜔 formulation is implemented in the inner region of the boundary layer and it is substituted by 𝜅−𝜀 model for the free shear region of the ﬂow. In the 𝜅−𝜔−𝑆𝑆𝑇 model, the turbulent kinetic energy 𝜅 and the speciﬁc rate of dissipation 𝜔 are expressed through the following differential equations: 𝜕𝑘 𝜕𝑡+𝑈𝑗𝜕𝑘 𝜕𝑥𝑗=𝑃𝑘− 𝛽∗𝑘𝜔+𝜕 𝜕𝑥𝑗[(𝜈+𝜎𝑘𝜈𝑇)𝜕𝑘 𝜕𝑥𝑗] (29) 𝜕𝜔 𝜕𝑡+𝑈𝑗𝜕𝜔 𝜕𝑥𝑗=𝛼𝑆2−𝛽𝜔2+𝜕 𝜕𝑥𝑗[(𝜈+𝜎𝜔𝜈𝑇)𝜕𝜔 𝜕𝑥𝑗]+2(1−𝐹1)𝜎𝜔21 𝜔 𝜕𝑘 𝜕𝑥𝑖 𝜕𝜔 𝜕𝑥𝑖 where the eddy viscosity, 𝜈𝑇 is expressed as : 𝜈𝑇=𝑎1𝑘 max (𝑎1𝜔,𝑆𝐹2) (30) while 𝐹2=𝑡𝑎𝑛ℎ([2√𝑘 𝛽∗𝜔𝑦,500 𝑣 𝑦2𝜔]2 ) (31) 𝑃𝑘=𝑚𝑖𝑛 (𝜏𝑖𝑗𝜕𝑈𝑖 𝜕𝑥𝑗,10𝛽∗𝑘𝜔) (32) 𝐹1=tanh ([𝑚𝑖𝑛 (𝑚𝑎𝑥 [√𝑘 𝛽∗𝜔𝑦2,500 𝜈 𝑦2𝜔],4𝜎𝜔2𝑘 𝐶𝐷𝑘𝜔𝑦2)]4 ) (33) 𝐶𝐷𝑘𝜔=max (2𝜌𝜎𝜔21 𝜔 𝜕𝑘 𝜕𝑥𝑖𝜕𝜔 𝜕𝑥𝑖,10−10) (34) The closure constants involved in the 𝜅−𝜔−𝑆𝑆𝑇 model are shown in Table 2. The following boundary conditions for 𝜅 and 𝜔 are set on the wall: 𝑘𝑤𝑎𝑙𝑙 =0 and 𝜔𝑤𝑎𝑙𝑙 =106𝑣 𝛽(∆𝑑1)2 Table 2. Closure constants of the k – ω – SST model. 𝜅−𝜔 𝜅−𝜀 𝑆𝑆𝑇 𝜎𝑘1=0.85 𝜎𝑘2=1.00 𝛽∗=0.09 𝜎𝜔1=0.65 𝜎𝜔2=0.856 𝛼1=0.31 𝛽1=0.075 𝛽2=0.0828 2.4.3 Boundary Conditions Periodic boundary conditions are applied when the ﬂuid across two opposite planes of the computational domain are the same [52]. For a fully -developed ﬂow, it is not necessary to simulate the full length of the pipe so, to reduce the number of cells and the computational time, the domain simply comprised a small section of the pipe and cyclic or periodic boundary conditions were applied, meaning that the results of the outlet were the new input of the inlet. No slip -boundary conditions were applied on the pipe wall to reproduce the effects of friction with null velocity at the wall and a static pressure boundary condition of zero pascals was set at the outlet. 2.4.4 Discretization The discretization of the partial derivative equations at the grid points use the Finite Volume Method (FVM) . A second order discretization scheme for evaluat ing the convective and diffusive terms were selected to ensure that the solution remain bounded because it has been proven that higher order discretization schemes bring convergence problems and instability of the solution [56]. 2.4.5 Pressure -Velocity Co upling Scheme The Semi Implicit Method for Pressure Linked Equations (SIMPLE) scheme, which is broadly used in CFD simulations [53], is the one employed in the present study. There is not an equation for controlling the evolution of the pressure in the se t of the Navier -Stokes equations and, for this reason, an iterative procedure is needed to relate the pressure and velocity; one disadvantage of the SIMPLE scheme is that it takes more time to converge. 2.4.6 Convergence Criteria Due to the presence of non -linear terms (convective) in the momentum equations, convergence criteria in CFD needs to be carried out by an iterative method. A residual value was used to monitor convergence and to assess proximity to the solution. The iterations should stop when the difference between the estimated solution and the previous approximation has reached a maximum residual value [54]. The maximum residual value settled for the following simulations are 1.0 E -6. 3 Results and Discussion The results of the experimental and computational techniques are presented and discussed in this section. First, experimental results are presented where the attention is restricted to the velocity proﬁles and the shear stresses distribution inside the pipe. Subsequently , the computational results are presented and compared with the experimental data. 3.1 Experimental Results 3.1.1 Velocity Proﬁle Analysis Fig. 2a presents the normalised velocities over the normalised distance from the wall in linear axes. In order to best illustrate the velocity distribution across the pipe, the normalised velocity 𝑈 𝑈𝑚 is plotted against 1−𝑟 𝑅, with the maximum velocity naturally obtained at the centre line, 𝑈𝑚=8.69𝑚/𝑠 . As discussed previously, in t his experiment, the Reynolds number was found to be very high ( 𝑅𝑒=57469 >> 4000 ), validating the assumption of a turbulent regime. Fig. 2a shows the turbulent region, excluding the viscous sub -layer and its approximation by a parameter - ﬁtted power -law cu rve. It illustrates that the power -law exponent n was found to be 7 .48, thus deviating by approximately 7% from the one -seventh power -law rule (equation 9). It also demonstrates that the total velocity corresponds to ≈68% of the centreline velocity. In Fig . 2b, the three characteristic regions of the turbulent boundary layer can be distinguished as viscous sub-layer, buffer layer and log -law layer, informed by a wider theoretical background on the turbulent ﬂow ﬁeld regions in the relevant literature [13,24,55]. The ﬂow in the viscous sublayer is very close to the tube wall and it is laminar, 0≤𝑦+≤5. In the buffer layer the turbulence is progressively rising, 5≤𝑦+≤30, and the ﬂow in the turbulent or log -law layer is turbulent, 𝑦+≥30. (a) Approximation of the normalised velocity ﬁtted with the power -law curve. (b) Dimensionless velocity proﬁle showing the regions of the turbulent boundary layer. Fig. 2: Experimental velocity proﬁles. Fig. 3: Theoretical and experimental shear stress across the pipe. 3.1.2 Wall Shear Stress Analysis Static pressure drop The wall shear stress related to the pressure drop over a known length of the pipe was calculated by the pressure difference in 𝑚𝑚 𝐻2𝑂. The value obtained from an analogue manometer is 2.7𝑚𝑚 𝐻2𝑂 for a length of 3.81𝑚. Using equation (11), the wall shear stress measured by this method is found to be: 𝜏𝑤=0.182 𝑃𝑎 Mean Square Signal (45° Hot -wire) The shear stress at the wall in this method was found by extrapolating the best ﬁtted line of the experimental results. The linear best -ﬁt equation, shown in Fig. 3, was used to ﬁnd the value of the wall shear stress when 𝑟= 0. Measurements of the RMS voltage have been taken at 10 mm increments along the diameter of the pipe. The Reynolds shear stresses were obtained through equations (14) and (15) and the linear ﬁt shear stress was computed by equation (16). Thus, the wall shea r stress using the mean square signal of a hot -wire is estimated as: 𝜏𝑤=0.162 𝑃𝑎 Preston Tube The wall shear stress was measured using a Preston tube, positioned so that its outer surface was tangential to the pipe wall, while facing upstream. An appropriate conﬁguration of the analogue manometer enabled the dynamic pressure to be read. The pressure reading was 𝛿𝑃=2.7𝑚𝑚 𝐻2𝑂. Using equations (18) to (21), 𝑋∗=1.890 and 𝑌∗=1.927 were obtained. Finally, equation (22) resulted in the following wall shear stress: 𝜏𝑤=0.188 𝑃𝑎 Clauser Plot Fig. 4: Semi -logarithmic plot with different values of 𝐶𝑓 from Sherman [44]. Fig. 5: Dimensionless velocity over dimensionless distance from the wall. Fig. 4 represents the Clauser plot, illustrating that the line of best ﬁt line is nearly aligned with the constant 𝐶𝑓/2 line at 0 .002. Thus, the wall shear stress was calculated from equation (28), giving a wall shear stress value of: 𝜏𝑤=0.178 𝑃𝑎 Fig. 5 depicts the variation of the dimensionless velocity 𝑈+ against the dimensionless distance 𝑌+, obtained from the data of the mean -square signal from a hot -wire. This graph is in agreement with the theory of the law of the wall or Clauser plot (Fig. 2b). 3.1.3 Error of Measurement The margin of error in this work is reported to be within 2% in the hot -wire measurements of the mean velocity ﬂuctuations using a CTA with linearizer [35]. The pressure error is the same order as the experimental uncertainty due to the difﬁcult nature of turbulent ﬂow over an oriﬁce. The errors on hot -wire positioning and velocity measurements are 25𝜇𝑚 and 0.25𝑚/𝑠 respectively. Data points are accepted if they are between two to three times the reference proﬁle of the error bars; if they are beyond this value, they are inconsistent. In this study, the data manages a close agreement with the reference solution, the mean pressure -gradient method, and all the experimental points obtained are consistent. 3.2 Comput ational Results Analysis of different aspects of turbulence modelling, mesh independency and periodic boundary conditions helped to deﬁne the optimal parameters for a CFD simulation. The CFD simulation is validated against the data points obtained from th e experiment using the mean square signal from a hot -wire method. One of the most important aspects to analyse is the sensitivity to the grid. The sensitivity test permits the CFD user to analyse the effects of ﬂuid behaviour with the variation of a speciﬁ c parameter in the simulation. A mesh independency study was carried out to examine the appropriate mesh to give an approximation of numerical error in the CFD simulation. In the present study, four signiﬁcantly different grid resolutions were simulated and compared, producing remarkable data, shown in Table 3. Table 3: Comparative table of mesh independency tests. Turbulence model k-ω SST Discreti zation Scheme Second -order upwind Convergence Criteria 10-6 P-V Coupling Scheme SIMPLE Scheme Boundary Conditions - No-slip in the walls - Periodic inlet/outlet Grid method 3D Multiblock - Hexahedral mesh Number of cells 3060 (coarse) 9520 (medium) 30240 (fine) 90045 (very fine) Iterations to Converge 13589 4510 12914 26625 Maximum Wall Y+ 0.053 0.028 0.028 0.025 CPU User time /Parallel*4 1463.88 s 764.764 s 4890.82s 55457.7 s Wall clock 126.53 s 99.75 s 1016.62 s 13339.77 s Mbytes used cells 3 9 29 77 Virtual Memory Usage 0.258 GB 0.279 GB 0.366 GB 0.414 GB For the four different types of mesh simulated (Table 1), the medium mesh, 9520 cells, gave a good resolution that only differed 1% from the ﬁnest mesh and exhibited the same level of accuracy but with far lower computational times and memory usage. While it is generally assumed that signiﬁcant improvements are always attained when reﬁning the mesh [57], the results demonstrate that this is not always necessarily the case; with 9520 cells, the result is mesh independent. It is worth noth ing that, despite the beneﬁts of hexahedral cells, even this non -orthogonal cell can reduce the numerical stability of the solution and it can also produce a slower convergence [56]. A post -processing tool was used to generate the contour of velocity of the medium mesh with periodic boundary conditions and 𝜅−𝜔−𝑆𝑇𝑇 turbulence model. Measured at the outlet of the pipe, the contour shows that the lowest velocities can be found at the pipe wall while the highest velocities can be observed at the pipe centre and that the velocity progressively increases from 0𝑚/𝑠 for the former to 8.818 𝑚/𝑠 for the latter, Table 4. Table 4: Parameters for the simulation of the contour of velocity. Parameter Value Contour of velocity Pipe inner diameter 0.102 m Length 0.004 m Reynolds number 57469 Air density 1.19 kg/m3 Air viscosity 1.82×10-5 kg /ms Von-Karman constant 0.41 (±0.1) Mass flow rate 7.5×10-2 kg/s Velocity 7.5 m/s Nearest wall distance 3.43×10-5 m 3.3 Experimental vs. Computational Results 3.3.1 Velocity proﬁles comparison A comparison of the computational and experimental results for the velocity proﬁles across the pipe, presented on logarithmic axes in Fig. 6a for an easier assessment of the ﬂow near the boundaries, indicates a satisfactory agreement between the two methodologies, with higher discrepancies appearing near the wall boundary. Moreover, a comparison of the dimensionless velocity proﬁle across the pipe, shown in Fig. 6b, likewise denot es a good agreement between the experimental and computational results, albeit with some inconsistent values close to the wall due to the presence of the turbulence velocity. 3.3.2 Wall shear comparison As with the velocity proﬁle distribution, a compari son of the theoretical, experimental and computational results for the shear stress distribution across the pipe surface, as shown in Fig. 7, indicates a close agreement between the computational and experimental results. The CFD simulation produces almost identical results to the theoretical results close to the wall and near to the centreline, while small differences appear with the theoretical distribution in the region 1−𝑟 𝑅 ∈(0.61,0.10); this region coincides with the region of the pipe where the velocity ﬁeld deviates the most from the experimental results. (a) Normalised velocity versus the dimensionless distance from the pipe wall on logarithmic axes. (b) Dimensionless velocity U+ versus the dimensionless distance from the pipe wall Y+. Fig. 6: Velocity proﬁles comparison between experimental and CFD results . Fig. 7: Comparison of experimental, theoretical and CFD results of the shear stress distribution across the pipe. 3.4 Assessment of Errors As the only direct method, the static pressure drop method is taken as a reference for assessing the validity of each method presented herein for the estimation of the wall shear stresses caused by fully developed turbulence ﬂow. Table 4 displays a comparison between the exp erimental and computational results. As illustrated, the CFD wall shear stress returns the closest agreement with the reference solution, with an error of approximately 1.6% compared to the experimental data. It is of interest to note that the CFD results are positioned exactly between the theoretical and experimental results of the mean square signal from a hot -wire method. Table 5: Comparison of the experimental and computational methods for the calculation of wall shear stresses. Method Wall shear stress [Pa] Error Mean Pressure – Gradient method 0.182 Reference 3D CFD 0.179 +1.6% The law of the wall 0.178 +2.2% Preston tube 0.188 +3.3% Mean square signal from a hot -wire 0.162 +10.4% 4 Conclusion This paper has discussed the velocity proﬁle and the mean shear stress in fully -developed turbulent pipe ﬂow using air as working ﬂuid and a Reynolds number of the order 50000. The ﬂow has been considered as incompressible. Measurements have been conducted with four different methods and their results are in broad agreement. These results were compared with axisymmetric RANS simulation results obtained with a commercial CFD software package in which a combined 𝜅−𝜀 / 𝜅−𝜔−𝑆𝑆𝑇 turbulence model was used. The model parameters and the initial and boundary conditions were given and the accuracy of the measurements were assessed by a comparison of the different approaches. This paper has demonstrated that, in all the experimental and computational methods tes ted here, the turbulent effects are well deﬁned where there is interaction between the ﬂuid and the wall. Importantly, it demonstrated that, while RANS is a slightly outdated approach that certainly does not capture all the ﬂow physics of the laboratory me asurements, for this particular case, with a high Reynolds number and utilising a standard computational turbulence model readily available with commercial packages, the numerical results produce a close agreement with the experiments. This demonstrates th at, for certain cases, CFD simulations could be used as an alternative or complementary methodology to the experiments. Acknowledgements The authors are grateful to Mr. Dennis Cooper and Dr Antonios Xenakis for their kind support and contributions.