A study on proximal region of low Reynolds confluent jets--Part 2: numerical prediction of the flow field.
One of Sweden's 16 national environmental qualities is good built environment: "Cities, towns, and building communities must provide a good living environment and contribute to improve the regional and global environment." The main objective for heating, ventilation, and air-conditioning (HVAC) systems in a building is to ensure a healthy and comfortable indoor climate, preferably at a low cost and with minimal energy usage and environmental impact. The building sector accounts for over 45% of the society's total energy use and contributes over 15% of the total C[O.sub.2] emissions. Built environment is the largest material-consuming industrial sector. The EU aims to reduce energy use and C[O.sub.2] emissions by 20% by 2020 and carbon emissions by 80% by 2050. One of the priority areas for achieving these goals is energy efficiency in buildings. Thus, the need to develop sustainable buildings and innovative HVAC system solutions to reduce primary energy use and power demand is very high.
Ventilation systems, thermal comfort, and air quality within the built environment are important issues as they are related to both energy conservation and the health of the occupants. Poor indoor environment conditions, e.g., in offices and classrooms, cost large amounts of money in health care, administration, and lost productivity. This stresses the importance of well-functioning HVAC systems in the built environment. The proposed project presents a new air distribution system with application of confluent jet ventilation. An energy efficient and an air-quality efficient ventilation system for industrial premises was introduced by Awbi (2003); Cho et al. (2008); and Janbakhsh et al. (2010), which is called "confluent jets." For a new design, engineers would need suitable and accurate three-dimensional turbulence models to predict the dynamics of multiple round jets (e.g., velocity components) in the proximal regions, which contain recirculation due to entrainment, occurrence of transition to turbulence, converging (deflection), merging, and combining.
The interaction (shear stress) between the flows within a jet and its boundary (the surrounding fluid), i.e., entrainment, plays the most important role in the development of a jet. The decay of axial velocity of a free jet and its outward spreading are natural consequences of entrainment, because the flow rate of entrained fluid by a laminar or turbulent jet increases with increasing axial location from the nozzle edge. Theoretical analysis (Schlichting and Gersten 2000) of entrainment has been experimentally verified by several investigators (e.g., Ricou and Spalding 1961, Crow and Champagne 1971, and Hill 1972). By introducing a jet within the boundary of another jet (e.g., twin jets), the entrainment phenomena will be different from those for free jets and would influence jet characteristics (Grandmaison et al. 1998). Free confluent jets are another example where the surroundings of one jet contain other jets.
When free round jets are issuing from different apertures with parallel axes in the same plane of flow, they coalesce or merge with each other and move as a single jet at a certain distance downstream (Tanaka 1970, 1974; Tanaka and Nakata 1975). This coalescence probably causes slower centerline decay of the issuing jets. Proximal regions of free round jets contain recirculation due to entrainment and transition to self-similar turbulence, which becomes more complicated for confluent round jets due to the existence of neighboring jets that cause converging, merging, and combining of multiple fluid streams. A study of the interaction between jets and mixing fields with strongly three-dimensional character provides knowledge of this complex flow, for which published data are meager. These confluent jets exist in different applications such as air supplies in ventilation systems (Cho et al. 2008, Janbakhshet al. 2010), air curtains, combustion nozzles, and pollutant dispersion and plume dilution from exhaust stacks with enough spacing. Cho et al. (2008) showed that wall confluent jets from a ventilation supply device penetrate to a longer distance compared to regular wall jets. The investigation by Janbakhsh et al. (2010) showed that the confluent-jet system produced better thermal comfort, which can work both for heating and cooling of industrial premises. The proximal region of low Reynolds number confluent jets needs to be studied in order to get more knowledge about flow behavior and to be able to optimize the supply device. Many comprehensive experimental, numerical, and analytical studies on parallel twin jets (dual jets), along with a row of jets, exist, for which reason review of some important aspects of these flows may be useful for the study of confluent (multiple) jets.
The results from review of twin jets also give some fundamental understanding about multiple jets. According to the authors' knowledge, most of the studies on dual jets were carried out on twin plane jets and only a few address twin round jets. Twin free jets usually consist of two coplanar identical round (or plane) nozzles of same diameter (or width) with parallel streamwise axes, which are separated from each other with certain spacing (S). The nozzle spacing (S) ratio can be used as a characteristic length for the twin jet configuration. A line of symmetry (or plane) bisects two jets and is the location toward which the jets converge. Twin jets pass an initial region right after they are issued from the nozzles. Then a sub-atmospheric static pressure is created between the jets due to mutual entrainment. This region is known as the "recirculation region" (Miller and Comings 1960, Tanaka 1970). The sub-atmospheric static pressure creates a free stagnation point and considerable reversal flow on the line (plane) of symmetry, while the twin jets deflect toward each other due to the increase of sub-atmospheric pressure in the converging region. The axis of the jets follows an arc of a circle and the deflection is due to the unsymmetrical mixing on each side of the jets. The outer edge of a curved jet experiences greater streamwise turbulence than the inner edge due to centrifugal force and deflection of jets (Harima et al. 2005, Tanaka 1970). The mean streamwise velocity is negative at the symmetry line (plane) until the inner shear layers of the two jets start to interact, which indicates the beginning of the so-called "merging region." In the merging region, the interactions between the two jets rise and the mean streamwise velocity at the symmetry line (plane) increases to a maximum value at the beginning of the combined region. The position of maximum velocity and maximum pressure of the combined jet are independent of Reynolds number and depend only on the geometrical pattern (Tanaka 1974). The super atmospheric static pressure redirects the merging jet streams into a single combined jet symmetrical about the symmetry line (plane) (Miller and Comings 1960, Tanaka 1970). The combined region starts right after the combined point where the twin jets completely combine and resemble a single free jet (Okamoto et al. 1985). It is shown that the main features of high-speed twinjet are quite similar to low-speed incompressible twin jets (Moustafa 1994), but by increasing Reynolds number or decreasing spacing between jets, turbulent energy and interference grows, i.e., the twinjets attract acutely (Yin et al. 2007). These five regions (initial, recirculation, converging, merging, and combined) are typical for twin jets, and depending on the design, some of them may appear together. For example, in the analysis of dual jet (from parallel slot nozzles) flow by Miller and Comings (1960) the flow region was only divided into jet convergence and combined jet.
Importance of Inlet Boundary Conditions for Free Jets
When the flow leaves the leading edge of a nozzle, it becomes unstable due to disturbances to the velocity profiles. Two length scales can characterize a considerably complicated round jet: Inlet boundary shear-layer thickness (e.g,. momentum thickness) and nozzle diameter. Near field largescale vortices depend on the inlet momentum thickness, while jet diameter is the controlling length scale afterward. While the inlet boundary conditions of a round jet affect the largescale vortices by forming axisymmetric vortex rings close to a nozzle exit and helical instabilities downstream, sometimes in a transition between two states (Dimotakis et al. 1983), the length for the flow becoming self-similar can be different for mean flow properties from higher order moments. Reviews of previous investigations of turbulent round free jet assess the effect of inlet boundary conditions on flow development. Gouldin et al. (1986) summarized the round jet data from previous experimental results and attributed the cause of variation in the spreading rate, virtual origin, and centerline mean velocity to the inlet boundary conditions. As a result, two strategies were suggested regarding the problem of choosing inlet boundary conditions: design a well-defined laminar inlet condition or turbulent inlet boundary layer. Transition modeling of large scales from laminar to turbulence is a difficulty for the first idea, while the requirement of extensive measurements (e.g., means, higher moments, and their correlations) for specifying the turbulent boundary is a difficulty for the second alternative. George and Davidson (2004) related production of asymptotic dependence of inlet boundary conditions to vorticity production, convection, and diffusion by giving some examples. The study by George and Davidson (2004) stated that single-point Reynolds averaged Navier-Stokes (RANS) turbulence models are not able to perform the asymptotic effects provided by the inlet boundary conditions. In contrast, Large Eddy Simulation (LES) overcomes this problem by resolving the large turbulence scales that are important for canonical flows. All these show that providing accurate inlet boundary conditions is important for the correct prediction of the initial and transitional regions. In the present paper, inlet velocity profiles, turbulent kinetic energies, and their specific dissipation rates for nozzles are used from a verified numerical simulation in the companion to this study (Ghahremanian and Moshfegh).
Anomaly in Modeling of Round Jets
Predicting the spreading of the fully developed region of round jets is a common problem with most turbulence models, and this shortcoming is known as the round-jet/plane-jet anomaly. Non-zero vortex stretching is responsible for the primary energy transfer mechanism from large to small eddies (Pope 1978). Therefore, dissipation must increase by enhancing vortex stretching as an energy transfer to smallest eddies, which is not suitable for RANS turbulence models. Pope (1978) proposed a non-dimensional measure of vortex stretching in the dissipation of dissipation term in the e equation. While the proposed vortex stretching parameter is zero for a plane jet, it is nonzero for axisymmetric mean flow and normally very small in axisymmetric boundary layers (because of large [omega]). There are some other modifications proposed by Spalding (1971), Launder et al. (1972), Morse (1977) and McGuirk, and Rodi (1977) but with no convincing physical explanations (Pope 1978). Cho and Chung (1992) considered the effect of entrainment on the round jet anomaly and proposed a three-equation model by including a transport equation for the intermittency of the standard k-[epsilon]. Ghirelli (2007) also derived a three-equation model by adding a transport equation for the age of turbulence to standard k-[epsilon] in order to overcome the mentioned anomaly. In the k-[omega] models derived by Wilcox (2006) and Menter (1994), the dimensionless vortex stretching parameter is considered in the context of [omega]-equation. Being aware of the round jet anomaly in the fully developed region, another question concerns how to simulate laminar, transition, and turbulent regions of free round confluent jets with a single turbulence model. On the other hand, a fairly rigorous test for both computational fluid dynamics and turbulence modeling is required to succeed in treating cases with strongly interacting free turbulent shear flows. The following summarizes the various methods that can be applied to predict all regions of round confluent jets. A. Direct Numerical Simulation (DNS) can obviously resolve all the flow scales, but is limited by computer resources. B. Large Eddy Simulation (LES) gives promising results for proximal regions, however, LES is sophisticated, costly, and more time consuming for engineers than Reynolds Averaged Navier-Stokes (RANS) models. Olsson and Fuchs (1996) briefly stated that, "Most turbulence models are not capable of modeling the transition from laminar to turbulent flow directly." Olsson and Fuchs (1996) also presented LES results on proximal region of round jet. C. If the transition points are known from DNS or experiments, it is possible to generate mesh for the separated laminar region and resolve it by RANS models; this is called the "indirect method." D. Another technique of employing RANS models for resolving mixing transition from laminar to turbulence is called "low Reynolds" turbulence approach, even though they are tuned for high Reynolds number flows. The low Reynolds Shear Stress Transport (SST) k-[omega] model (Menter 1994) is an example of low Reynolds RANS models. The last technique has been chosen for the present paper based on a previous study by the authors (Ghahremanian and Moshfegh 2011) in which this turbulence model showed good agreement for a single round turbulent jet.
In the present paper, results of three-dimensional modeling of isothermal, free, turbulent, round confluent jets issuing from a square array of six columns by six rows of nozzles (with identical spacing) placed on a cylinder with two-equation turbulence model (SST k-[omega] with low-Reynolds correction) with resolved inlet profiles are compared and verified with hot-wire anemometry.
The size of a well-insulated test room is 4.2 (length) x 3.6 (width) x 2.5 (height) [m.sup.3] (165.35 x 141.73 x 98.42 [in.sup.3]), where the temperature can be controlled within [+ or -] 0.1[degrees]C (32.18[degrees]F). The test rig is located inside an operating room that is insulated from the main laboratory ambient air. The wooden test room walls include a water pipe system for temperature control. The airflow is supplied from a centrifugal fan with a straight radial blade. A frequency regulator controlled the blower power in order to get the desired airflow rate. The supply air duct is connected to the fan and an orifice flowmeter. Then a flexible-rubber duct is connected to a straight 2 m (78.74 in.) long aluminum pipe to absorb mechanical vibrations. The vertical supply device with a diameter of 0.125 m (4.92 in.) is placed at the end of the 2 m (78.74 in.) long pipe. Confluent jets issue into a large test room with air that is initially at rest. It is assumed that the confinement by walls has no effect on jet development and merging. No filters or honeycomb were used for this measurement. The air exits the test room freely through a rectangular exhaust on one of the sidewalls. Conducting experiments in a confined environment, i.e., a small enclosure, may bring a reduction in the momentum of round confluent jets (Hussein et al. 1994). Table 1 compares test room facilities and shows that relative magnitudes of both ratios for the present test room are sufficiently large, i.e., elimination of the effects of room enclosure.
The readings from two different hot-wire measuring techniques (standard hot-wire [HWA] and flying hot-wire [FHW]) were compared with those from a laser Doppler anemometer (LDA) by Hussein et al. (1994) and Capp et al. (1990) for a high Reynolds, axisymmetric, turbulent, round, free jet for all moments up to third order. They showed a substantial difference between hot-wire techniques and LDA in places with high local turbulence intensity. The differences are consistent with reported hot-wire uncertainties such as cross flow, rectification and drop-out, due to the presence of return flow in the test place and high turbulence intensity. Second-order differential and integral terms of momentum equations were satisfied with the results from measurement by FHW and LDA but not by HWA. Velocities ranging from a few centimeters per second to about 10 m/s (32.8 ft/s) were measured for the present paper that represent a low Reynolds number flow because of the small diameter of nozzles. Movement of large structure and hardware for LDA or FHW (compared to a single hot-wire probe) within the test room would have caused large disturbances and parasitic air motions at a low Reynolds number, flow-sensitive confluent jets. Placement of large hardware might also cause an important interference in the flow field. Velocities and their root mean square (RMS) can easily be measured by single probe hot-wire.
Following these comparisons, a 55P11 constant-temperature hot-wire anemometric bridge was applied to measure velocity and its RMS. The straight single-sensor miniature wire with especially long prongs (5 mm or 0.2 in.) is made by DANTEC and one-dimensional platinum-plated tungsten probe has 5 [micro]m (0.0002 in.) diameter and is 1.25 mm (0.05 in.) long. The general-purpose probe was operated at a temperature of 200[degrees]C (392[degrees]F) with an over-heat ratio of approximately 0.64. Calibration of a hot-wire probe was carried out by a low-speed and high-speed open loop wind tunnel that is located in a temperature-controlled calibration room. Fourth-order polynomial least square regression was used to evaluate the velocity calibration data. Data acquisitions were performed with a National Instrument NI USB-6215.
A temperature sensor was placed in the vicinity of the anemometer probe to ensure that isothermal conditions were fulfilled. The position of the temperature sensor did not disturb the flow and was not influenced by the heat from the hot-wire probe. Several thermocouples were installed on walls, the ceiling, the floor, exhaust, and at different heights of the test room center to control temperature of test rig.
A LabVIEW computer program was employed for highly automated data gathering considering the high volume of data and long acquisition times. A traversing system with uncertainty of 0.125 mm (0.00492 in.) was applied for automated movement of the hot-wire probe and temperature sensor. The three-dimensional traversing system was also controlled by the LabVIEW program.
Nozzles of the confluent jets with exit diameter ([d.sub.0]) 0.0058 m (0.23 in.) are located on a cylindrical air supply (Figure 3), which is placed at the center of the longest wall of the test room. The nozzle exits were designed so that their inlet profiles have saddle-back shapes and differ from top hat-like velocity profiles (Ghahremanian and Moshfegh). A nozzle contraction with small aspect ratios cause an increase in the centerline velocity of confluent jets and a decrease in the velocity on the confluent jet borders in the downstream region of the nozzles. Entrainment of nominally static fluid surrounding the nozzle exit can also amplify the contraction effect with an increase in radial momentum.
The axial component of instantaneous velocity in a cross-section (z axis) of Column 4 was measured starting from the center of nozzle at the third row from top and fourth column from left (3R-4C) at Reynolds number 3290 by using a single-wire sensor (Figure 10). Every set of measurements includes a series of data along the nozzles cross section in Column 4 at different distances ranging from 0.34 [d.sub.0] to 20 [d.sub.0] downstream from the exit sections with steps of 0.17 [d.sub.0]. Each measurement covered the complete profiles of three and half jets and finally shifted up the sensor to the next step, when the measured value after nozzle 6R-4C reached 3% of the maximum velocity. After cross-sectional measurements, the axial component of instantaneous velocity was recorded along the geometrical centerline of nozzles 3R-4C, 6R-4C, 3R-6C, and 6R-6C with steps of 0.17 [d.sub.0]. Measurements were repeated for the centerline of nozzle 6R-4C and the recorded data matched together (Figure 1). Every data point was taken at 5 kHz for 180 seconds. After all measurements, the mean axial velocity and its root-mean-square (RMS) were obtained by averaging all the instantaneous samples.
Measured turbulence levels during this work were less than 35%, which Bruun (1995) showed that the problem of rectification is small below this limit, considering homogeneous and isotropic turbulence. The relative expanded uncertainty of velocity measurements by the single-sensor hot-wire probe in air is 4.23%. Sources of uncertainty and their relative standard uncertainty can be summarized as follows: calibration (2%), linearization (curve fitting) (0.5%), analogue/digital board resolution (0.13%), probe positioning ([approximately equal to] 0), air and sensor temperature variations (1%), ambient pressure variations (0.6%), and humidity ([approximately equal to] 0).
COMPUTATIONAL SETUP AND NUMERICAL SCHEME
Geometrical Setup and Boundary Conditions
The research finite-volume solver Fluent 13 was employed to numerically simulate the airflow of half of an array of 36 round jets, i.e., three columns of six jets. Half of six columns, which is split by a symmetry plane (Figure 2), has been modeled in order to reduce computational complexity, time, and cost of simulation.
Computational domain (Figure 2) starts from a curved solid wall (at a radius of 0.0625 m or 2.46 in.), D/[d.sub.0] = 21.55, and contains 18 nozzles. Then, it radially extends to pressure outlet faces at 100 [d.sub.0]. The longest arc length between pressure outlet and symmetry plane is chosen as 261 [d.sub.0], which is equal to an angle of 135[degrees]. While the angle between the center-tocenter of two columns of jets is 15[degrees], the total angle between the symmetry plane of the array and Column 6 becomes 37.5[degrees], therefore a 135[degrees] angle is more than a right angle and is wide enough to place pressure outlet. The geometrical details and fluid properties are summarized in Table 2.
In order to examine mesh independency, the model has been solved with three different scaled grid densities, i.e., 208, 576, and 1280 hexahedral cells per each nozzle exit plane. The computational grid of the nozzle's exit planes is illustrated in Figure 3. Identical cell distribution is used in three test cases: the greater the number of cells applied, the finer the resolution near the walls. The predicted velocity profiles at the outlet of nozzles were compared with measurement results. The results revealed that the grid density 208 is unsatisfactory and the case with 1280 cells produces almost the same information as a grid density of 576. Therefore, results for numerical prediction using 576 cells in the nozzle's outlet plane are presented in this paper.
The computational staggered grid is generated with hexahedral cells so that the total number used is 12 million cells for the selected case for grid independency. The mesh is refined enough near the solid wall, where the wall nearest [y.sup.+] is kept below one to solve all boundary layers compatible with low Reynolds correction. Special attention has also been paid to control the dimensionless streamwise, normal and span-wise spacing in order to avoid the generation of cells with high aspect ratio, which would degrade the numerical accuracy. The model decomposition and structured mesh is generated with Gambit 2.4.6.
The matrix of nozzles is respectively labeled by row and column numbers (Figure 4), i.e., 1R-4C stands for the nozzle from first row and fourth column. In this configuration, three different types of confluent jets can be seen based on the number of surrounding jets: jets at side rows or columns with five neighboring jets (Side Jets [SJ] e.g., 1R-4C, 4R-6C and 6R-4C), jets at corners with three neighboring jets (Corner Jets [CoJ], e.g. 1R-6C and 6R-6C), and jets in the middle of the matrix with eight neighboring jets (fully Confluent Jets [CJ] e.g. 4R-4C). Nozzle axes in columns are coplanar and parallel in the downstream direction, while nozzle axes in rows are coplanar without intersecting in the downstream direction. The nozzle axes in rows are obliquely aimed away from each other and form a diverging configuration. Despite the angle of 15[degrees] between the axes in rows, the length (or arc length) of spacing between nozzle axes in rows and columns was designed identical.
The supply device is simulated with the Realizable R k-[epsilon] model (R k-[epsilon]) and the Reynolds stress model (RSM) in order to provide the inlet profile for the confluent jets (see Ghahremanian and Moshfegh 2014). The simulated supply device in part I of this study (Ghahremanian and Moshfegh 2014) has the identical nozzle diameter (d0) and mesh resolution with the current model. Axial velocity profile of the nozzle outlets and its associated turbulence intensity (Figure 4) are verified with hot-wire anemometry in another part of this study by the authors (Ghahremanian and Moshfegh 2014). The velocity components, turbulent kinetic energy (k), and the specific dissipation rate ([omega]) at nozzle outlets are employed from the related work as the inlet profiles in the current model.
The mean axial velocity profile at the nozzle plane has two overshoots (Figure 4), which is similar to flow from an orifice nozzle studied by Mi et al. (2001a), Mi et al, 2001b, and Deo et al. (2007). Uneven peaks in velocity profiles appear at the nozzle plane in the z direction of two first rows, which is influenced by the flow direction in the main pipe. This unevenness is smoothed for nozzles at the bottom rows (4, 5, and 6). The faster convergence of the velocity profile in z direction than in the x direction can also be seen in all nozzles. Most of the presented velocity profiles tend to follow a top-hat profile after a few diameters downstream. Details of the above-mentioned behavior of the nozzle profile have been shown in another study by the authors (Ghahremanian and Moshfegh 2014).
The isothermal (no buoyancy effect) airflow is governed by the conservation laws of mass and momentum. Based on the above assumptions, the averaged continuity and Reynolds equations for steady-state, three-dimensional, incompressible and turbulent conditions are given by 9U.
[[partial derivative][U.sub.i]/[partial derivative][x.sub.j]] = 0. (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where [bar.[u'.sub.i]-[u'.sub.j]] are the second moment of statistical correlations called Reynolds stresses and should be modeled in order to close the system of equations. A common approach to model Reynolds stresses of Equation 2 is applying the Boussinesq hypothesis (Hinze 1975), which relates them to mean velocity gradients as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
The Boussinesq hypothesis is employed in the low Reynolds number Shear Stress Transport (SST) k-[omega] model. Only two additional transport equations for the turbulence kinetic energy, k, and turbulence frequency (sometimes called specific dissipation rate), [omega], need to be solved. Therefore, [v.sub.t] is assumed as an isotropic scalar quantity and computed as a function of k and [omega]. This approach results in relatively low computational cost associated with the modeling of turbulent viscosity ([v.sub.t]) compared to the alternative approach to solve six transport equations for each of the terms of the Reynolds stresses, plus additional scale determining equation.
Menter (1994) developed shear-stress transport (SST) k-[omega] by blending the robust formulation of the standard k-[omega] model (Wilcox 2006) for free shear flows (e.g., round jet) in the near-wall region with the free stream independence of the transformed k-[omega] in the far field. The transport equations for k in the SST k-[omega] model are given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
The turbulent Prandtl numbers ([[sigma].sub.k] and [[sigma].sub.[omega]]) for k and [omega] are respectively used in effective diffusivities (y is the distance to the next surface and [D.sup.+.sub.[omega]] is the positive portion of the cross-diffusion term). There are blending functions in turbulent Prandtl numbers that tend to one in the near wall region that activates the standard k-[omega] model and become zero away from surface that activates the transformed k-[epsilon] model. The term [[??].sub.k] describes the production of turbulence kinetic energy.
The second term in the right-hand side of the [omega] transport equation represents the production of specific dissipation. For the incompressible form, the third term of the right-hand side of Equation 5 represents the dissipation of [omega]. Transformation of the standard k-[epsilon] model into equations based on k and c leads to the introduction of a cross-diffusion term [D.sub.[omega]].
The turbulent viscosity, [v.sub.t], is defined as follows, where S = [square root of (2[S.sub.ij][S.sub.ij])] is the strain rate magnitude:
[v.sub.t] = [k/[omega]] [f.sub.SST] (6)
There is one coefficient in [f.sub.SST] that damps the turbulent viscosity and causes low Reynolds number correction, which is written as below:
[[alpha].sup.*] = (0.024 + k/6v[omega])/1 + k/(6v[omega])) (7)
In shear-layer edge of free shear flows, introduction of blending functions in cross diffusion term improves production of specific dissipation, [omega], which in turn enhances dissipation of turbulent kinetic energy, k, i.e., net production of k is reduced. Although vortex stretching is very small in axisymmetric boundary layers, vortex stretching is the reason for the energy transfer from large to small eddies and is nonzero for an axisymmetric mean flow, which corresponds to radial stretch of vortex rings.
The research finite volume solver (Fluent 13) was employed to numerically simulate the air flow from round confluent jets. The governing transport equations are solved with segregated memory efficient algorithm (Table 3). The double precision, steady state and pressure based solver is used for SST k-co model. Regarding discretization, the nonlinear terms are calculated with the second order upwind scheme and the viscous terms with the second order central scheme. The under-relaxation factor for pressure, momentum, turbulent kinetic energy, and turbulent dissipation rate (specific dissipation rate) is set to 0.3, 0.7, 0.5, and 0.5, respectively. The SIMPLE algorithm was used to solve the pressure-velocity coupling. If the sum of absolute normalized residuals of transport equations in the turbulent models for all of the cells in flow domain is below [10.sup.-6], the solutions were considered converged.
The calculations were conducted on a high-performance cluster with two processors that each consist of four cores and 32 GB system memory.
Measured and predicted axial velocity (i.e., velocity component in y-direction V) at the center-plane of Column 4 (nozzles: 3R-4C, 4R-4C, 5R-4C, 6R-4C) for Re = 3290 is illustrated in Figure 5. The SJs are strongly deflected toward their neighboring jets, but they behave muchlike free jets close to the nozzle exit. Contrary to the idea of SJs and twin jets, CJs do not bend toward the symmetrical planes in between them. The virtual origins of all the jets lie on the nozzle axes and at the same distance from the exit plane as if each jet were issuing in isolation. One notable feature about SJs is that they are completely displaced away from the nozzle axes toward neighboring jets. The reason is that each jet has a tendency to entrain fluid from its surroundings until they effectively merge. After the combined region, the SJs are fully mixed with their neighboring jets, and they do not independently exist. The static pressure for confluent jets recovers to the ambient pressure faster than side jets and corner jets.
Geometrical Centerline Decay
Figure 6 shows the decay of axial velocity along the geometrical centerline of nozzle 4R-4C. The centerline velocity of confluent jets is over-predicted by the SST k-[omega] in the transition region, while it matches with measured data in the initial and fully developed regions. This over-prediction, which is also reported by Leschziner and Rodi (1981) and Lai and Nasr (1998), could be because of excess turbulent kinetic energy, transitional phenomena, or lack of isotropic, and homogeneous turbulence. For this study, other turbulence models are also employed for the prediction such as the standard k-[epsilon], which could not predict the transitional region at all due to the familiar round-jet anomaly. The excess turbulent kinetic energy, lack of dissipation rates, and curvature for secondary strain effect in standard k-[epsilon] model forced the jets to become fully turbulent without going through any transition.
Measured and predicted axial velocity in the geometrical centerline of CJs (3R-4C), SJs (3R-6C, 6R-4C), and CoJs (6R6C) are presented in Figure 6. The centerline axial velocity of the CJs is close to that for a single jet (Ghahremanian and Moshfegh 2011) at y/[d.sub.0] [less than or equal to] 5. The nonparallel exit configuration brings entrainment of nominally static fluid surrounding the nozzle's exit into transitional regions of jet development with radial momentum and contribution to concentration of streamwise momentum in the centerline of the jet. The greater spread of confluent jets at the combined region compared to a single jet is due to an increase of static pressure rather than momentum diffusion (Tanaka 1974). SJs and CoJs rapidly decay to less than 4% of the maximum axial velocity at the geometrical centerline after their similar behavior to confluent jets down to end of core region (y/[d.sub.0] [less than or equal to] 1).
Confluent Jet Decay
The comparisons for velocity distribution at different cross-section distances from the nozzles in Column 4 is shown in Figure 7. Figure 7 shows that the predicted spreading of half of Column 4 at different distances from the nozzle's edge is consistent with measurement. Segregated jets have their own identity and characteristics after issuing from nozzles. The confluent jets begin to interact and the velocity in between jet areas increases gradually and then the velocity at the mid-point between jets becomes equal to that at the central area of the jets further downstream. Contrary to twin jets, there is no shift in the position of maximum velocity of CJs toward the symmetry line between them, i.e., CJs radially spread straight until their boundaries merge together. The velocity profile of the SJ becomes narrower due to greater outer shear flow, while CJs have top hat-like profile. As mentioned above, a faster decay and deflection of the SJ due to the lateral shear stress is also obvious in Figure 7. The maximum velocity point of the SJ in Figure 7 moves toward CJs until they merge and totally combine with its adjacent jet. The velocity profile for CJs and SJs including areas of interface shows a similarity for these three different low Reynolds number in initial (y/[d.sub.0] = 2.4, 4.5), merging (y/[d.sub.0] = 12.1) and combined (y/[d.sub.0] = 20.4) regions except in the converging region. Measurement and numerical simulation are perfectly matched in the initial region of confluent jets, which implies that the inlet boundary conditions employed were correct. As the confluent jets develop, the difference between measurement and numerical prediction increases and the maximum mismatch occurs in the merging region (y/[d.sub.0] [approximately equal to] 12). When the confluent jets completely merge and combine, the recorded experimental data become consistent with numerical modeling either in the magnitude of velocity or in spreading rate.
The static pressure distribution of center-plane of Column 4 is shown in Figure 8. All jets have positive static pressure after the nozzle exit in their core region and negative static pressure in between them. The static pressure reduces to a negative value for all jets until they merge and combine in downstream, which leads to a flat positive static pressure for combined jets. The greater sub-atmospheric static pressure between SJs or CoJs and their neighbor's jets accounts for the observable deflection (Figure 5 and Figure 10) and convergence. Then the super-atmospheric static pressure redirects the combined jet into a plane jet symmetrical about the center of matrix of jets. These two types of jets behave more similar to twin jets in terms of deflection, converging, merging, and combining to their neighboring jets except the position of merging point and combined point, which is not located in symmetry plane between jets. Interactions between SJs and their adjacent jets are stronger, therefore SJs at rows (columns) 1 and 6 deflect and combine with their adjacent row (column).
In Figure 9, comparisons between measurement and numerical predictions for the RMS of axial velocity at two positions are depicted. High turbulence levels are produced in the interacting region between jets, which effectively overshadows any effects of upstream history. The direction for the approach of the RMS axial velocity towards equilibrium is worth mentioning. As can be seen in a study by the authors for a single free round jet (Ghahremanian and Moshfegh 2011) and in Figure 9, the RMS velocity increases with downstream distance, which shows the production of turbulent kinetic energies, in excess of its dissipation until it reaches equilibrium. In confluent jets, a super-equilibrium level of turbulent kinetic energy remains in the interaction zones, in which this super-equilibrium decays during the approach to equilibrium.
The mapping of the velocity field of the interacting jets in a multi-lateral symmetry plane containing nozzle axes in the fourth and sixth columns and rows is depicted in Figures 10 and 11. A three-dimensional characteristic of the flow field can be seen in these two figures. A resulting single jet is expected downstream of the combined zone (after merging of jets) that has a tendency toward an equilibrium structure in respect to mean flow and turbulence properties. The diameter of the resulting single jet is smaller than the full length of the columns because of the converging lateral spread of the confluent jets, which is due to attachment of jets to each other and shearing stress between them. Jets in rows and columns have coplanar axes but their merging and combining are different due to parallel or diverging geometrical centerline. However, the present measurement was limited to the combined region and would have to be extended much further downstream in order to clearly see the exhibition of a single jet and its tendency to become self-similar (fully developed). The development of the turbulent kinetic energy for CJs is similar to a round single jet, i.e., laminar in the core region and highly turbulent in both shear layers. Since the SJs or CoJs deflect toward their neighboring jets, their turbulent kinetic energy also shows identical behavior to the velocity field. The outer shear layer of the SJs or CoJs contains the highest turbulent kinetic energy, which is due to great deflection.
The three Reynolds number (Re = 2125, 3290, and 4555) are based on the nozzle exit diameter ([d.sub.0]) and the bulk velocity (Ubuik) from the airflow meter (Table 2). These Re number were selected based on the total airflow that the test room needed and also the supply device could handle it without vibration. The idea behind the air supply device is low air velocity inside the diffuser and high velocity from the nozzles by employing small nozzles, which can reduce the energy usage by fan. Deflection, convergence, and combination of the maximum velocity line (centerline) of confluent jets with the three different Reynolds numbers studied is illustrated in Figure 12. [Z.sub.max] describes the vertical position of the maximum axial velocity for the three jets studied in Column 4. In the Column 4, [z.sub.max]/S = 0 means that the vertical position (z) of the maximum axial velocity ([V.sub.max]) of the jet lies on its geometrical centerline while [z.sub.max]/S = 1 represents that the vertical position (z) of the maximum axial velocity ([V.sub.max]) of the jet is completely deflected toward the geometrical centerline of its upper adjacent jet. In this paragraph, all results for Re = 3290 are discussed. An SJ comes out from the nozzle exit and the maximum velocity line follows its geometrical centerline up to 1.5 [d.sub.0] and then changes its direction toward its neighboring jet (5R-4C) until it reaches the adjacent jet after 4.2 S downstream, which is also shown in Figure 7. As discussed in the above sections, SJs (e.g. 6R-4C) converge and combine faster than the adjacent CJs (e.g., 5R-4C). CJs emanate from the nozzle exits to produce maximum velocity along the geometrical centerline in a very similar way to single free round jets, only with a different decay rate (Figure 6). As shown in Figure 12, the maximum velocity of the jet in between SJ and CJ (5R-4C) follows its geometrical centerline for a longer distance up to 4.5 S before starting to deflect toward its adjacent jet (4R-4C).
In Figure 12, Reynolds number dependency of an SJ (6R4C) and its adjacent jet (5R-4C) is clearly observable, while the maximum velocity of the CJ (4R-4C) follows its geometrical centerline for all three Re studied. It is worth mentioning that the combined point of an SJ (6R-4C) for Re 2125 occurs at a closer distance (3.67 S) to the nozzle lip compared to the other two Reynolds numbers (4.28 S). This means that SJs at lower velocity deflect more and combine faster with their adjacent jets. On the contrary, it seems that the deflection and amalgamation of the jet 5R-4C between SJ (6R-4C) and CJ (4R-4C) at Re 2125 and 3290 follows almost a similar pattern as for Re 4555.
In this study, confluent jets were found to have a complex and three-dimensional flow characteristics. Comparison of hotwire anemometry measurements and numerical predictions with turbulence model for 18 confluent jets having identical nozzle diameter, nozzle shape, and spacing have been carried out for three different Reynolds numbers. The results discussed above show a consistency in the whole three-dimensional flow field between measurements and predictions using the SST k-[omega] with low-Reynolds correction particularly in the initial and fully developed regions. Vortex stretching, which corresponds to radial stretch of vortex rings, is the reason for the energy transfer from large to small eddies (then dissipation) and is nonzero for the mean flow of round jets and is very small in the boundary layer regions of round jets. The reason for better accuracy of the low Reynolds SST k-[omega] model lies in the shear-layer edge of free shear flows, for which introduction of blending functions in cross-diffusion term improves the production of specific dissipation, [omega], which in turn enhances energy transfer and dissipation of turbulent kinetic energy, k, i.e., net production of k is reduced.
Applying modeled inlet boundary can also improve the prediction of confluent jets. An over-prediction of streamwise velocity is still present in transition or mixing region of fully confluent jets. The behavior of the mean streamwise velocities is found independent of nozzle's Re in the range studied except for the mixing region. Three different confluent jets exist in this investigation according to their interaction with neighboring flow. Apart from fully confluent jets (CJs), side jets (SJs) and corner jets (CoJs) behave differently; all of which have common initial, mixing, merging, and combined regions. SJs and CoJs deflect toward their adjacent jets and finally merge and combine with them, while CJs normally spread and amalgamate with each other. Lower local pressure is responsible for amalgamation of the CJs, but the static pressure reaches a minimum value between SJs and their neighboring jets, which results in the deflection of SJs.
The supply devices for ventilation systems based on the above-mentioned confluent jets may provide longer spreading of the air inside a premise due to merged and combined jets. Another advantage of using confluent jets is the high capacity of mixing with surrounding air in converging and merging regions. The most important benefit of using confluent jets in ventilation systems is the ability to work with both heating and cooling systems due to higher velocity. Despite the advantages of using confluent jets, the configuration studied is not optimal and needs further investigation. For example, the jets in rows have coplanar axes, but due to diverging angle from each other, the merging and combining of the jets occur at a greater distance compared to the jets in columns.
The authors gratefully acknowledge the financial support received from University of Gavle, Sweden. The authors are thankful for the assistance received by personnel at the Laboratory of Ventilation and Air Quality at the Center of Built Environment, University of Gavle, Sweden, and especially to Hans Lundstrom.
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Student Member ASHRAE
Shahriar Ghahremanian is a PhD student in the Department of Management and Engineering, Linkoping University and the Department of Building, Energy and Environmental Engineering, University of Gavle, Sweden. Bahram Moshfegh is head of the division of Energy Systems in the Department of Management and Engineering, Linkoping University and is also professor in the Department of Building, Energy and Environmental Engineering and Chair of Education and Research Board of University of Gavle, Sweden.
Table 1. Room Confinement Analysis Ratio of Room Ratio of Room Height to Equivalent Area to Diameter (6[d.sub.0]) Equivalent Area Present study (36 jets) 72 16000 Hussein et al. (1994) 200 50000 Malmstrom et al. (1997) 40-170 2000-30000 Nottage (1951) 36 4000 Table 2. Geometrical and Flow Details [d.sub.0] 2[r.sub.0] = 0.0058 L m (=0.23 in.) S (Spacing) 2.82 [d.sub.0] [rho] Height of nozzles 0.0023 m (0.09 in.) [mu] Contraction Ratio 4 Re = [[U.sub.bulk] x [d.sub.0]]/v [d.sub.0] 59 [d.sub.0] S (Spacing) 1.225 kg/[m.sup.3] (0.076 lb/[ft.sup.3]) Height of nozzles 1.7894 x [10.sup.-5] kg/ms (0.017894 cP)) Contraction Ratio 2125, 3290, 4555 Table 3. Numerical Scheme Grid Staggered Solver 3-dimensional, double precision, steady state, pressure-based Pressure-velocity Simple coupling scheme Spatial Discretization Gradient Least squares cell based Pressure Second order Momentum Second order upwind Turbulent Kinetic Energy Turbulent Dissipation Rate
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|Author:||Ghahremanian, Shahriar; Moshfegh, Bahram|
|Date:||Jan 1, 2014|
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