Numerical Study on Plane and Radial Wall Jets to Validate the 2D Assumption for an Idealized Downburst Outflow

School of Civil Engineering and Architecture, Chongqing University of Science and Technology, Chongqing 401331, China Chongqing Key Laboratory of Energy Engineering Mechanics & Disaster Prevention and Mitigation, Chongqing 401331, China China Construction Fourth Engineering Bureau Guizhou Investment and Construction Co., Ltd., Guiyang 550014, China School of Civil Engineering, Chongqing University, Chongqing 400045, China


Introduction
e occurrence of wall jet flows is common in many industrial applications. Traditional applications include ventilation, film cooling, and separation control over wings [1]. In some small-scale geometric engineering applications, turbulent jets are mainly used for heat transfer [2,3]. In civil engineering, the wall jet flow can be used to design laboratory (or numerical) simulations of large-scale downburst outflow, which is a high-intensity wind that results in failures of transmission lines [4,5].
ere are two main methods for generating wall jet flows depending on the angle of injection of the high-momentum fluid. If the high-momentum fluid is injected normally to the wall, the resulting flow field is classified as a radial wall jet [6], which is a logical way to achieve flow similarity of downburst outflow [7]. A plane wall jet is produced when the high-momentum fluid is parallel to the wall, and this can also be an idealized model for the outflow region of a downburst [8,9].
Both radial and plane wall jets have been extensively studied over the past few decades. Owing to the extensive applications of wall jets, there are many studies on them. Launder and Rodi [10] provided a comprehensive review that reflects the state-of-the-art experimental research conducted until 1980. Reviews on more recent literature can be found in Naqavi et al. [11] and van Hout et al. [12] for plane and radial wall jets, respectively. ese literature reviews suggest that the studies on wall jets usually focus on one type at a time. Radial [13][14][15] and plane [16][17][18][19][20] wall jets were investigated separately. Only a few studies have compared the basic characteristics, such as the evolution of the length and velocity scales, of the two types of wall jets. Tanaka and Tanaka [21] compared the velocity and length scales of their experimental radial wall jet with those available in the literature on plane wall jets. It was revealed that the evolution of the half-maximum velocity location is very similar, while there is a difference between the evolutions of the maximum velocities. Banyassady and Piomelli [22] obtained similar conclusions through a large eddy simulation. Guo et al. [23] identified that the confinement of an impinging jet has no significant effect on the velocity decay rate, and the presence of the upper confinement plate accelerates the wall jet growth rates compared to those reported in the previous plane and radial wall jet experiments. However, the numerical study results of radial wall jet by Fillingham and Novosselov [24] exhibited an excellent agreement with those of plane wall jet reported by Naqavi et al. [11] in terms of the evolution of both length and velocity scales. Bagherzadeh et al. [2] reported that wall roughness influences the decay rate. ey found that the decay of velocity increases with an increase in wall roughness.
Most previous studies on plane and radial wall jets focused on heat and mass transfers; they had relatively small Reynolds numbers [25,26] and were not suitable for the study of downbursts. Due to the influence of the wall, the free jet region of the impinging jets is also different from the turbulent round jet [27]. In applications related to civil engineering, the majority of the previous investigations on radial and plane wall jets focused on the profiles and time series of velocity [28][29][30]. It is useful to characterize the length and velocity scales for high Reynolds numbers. In addition, an external stream exists in most practical situations of a plane wall jet. e external stream also provides fluid for jet entrainment [31]. In the case of downburst outflow simulation, an external stream may be applied to simulate a translating event [32]. In addition, the results from different wall jet studies have apparent discrepancies even in some basic characteristics. ere is a fundamental geometric difference between plane and radial wall jets: radial wall jets have an additional direction of expansion. Although Lin et al. [33] verified that the frontal curvature has little effect on the resultant wind loading of a structure within a certain transverse width, which is a geometric analysis, the validity of the 2D assumption needs to be further investigated from the perspective of the difference between the longitudinal evolution of the 3D outflow and 2D wall jet. e primary objectives of the present study were (1) to perform a systematic parametric study through the numerical simulation of radial and plane wall jets to determine the characteristics of evolution and (2) to further investigate the 2D assumption for the downburst outflow. Following Section 1, Section 2 introduces the numerical simulation details of the radial and plane wall jets; Section 3 presents the comparison of the results from existing literatures with those predicted in the current simulation. e Reynolds-number dependency was investigated for both types of wall jets. Subsequently, the influence of the nozzle height from the plate on the radial wall jet and the effect of co-flow on the plane wall jet are presented. Based on the simulation results, the accuracy of approximating a downburst outflow with a plane wall jet was evaluated, as presented in Section 4. Section 5 summarizes the main findings of the present study.

Governing Equations.
A commercially available computational fluid dynamics (CFD) package, FLUENT, was used to simulate the impinging jet and plane wall jet. e conservation equations of mass and momentum for an incompressible fluid flow can be expressed as follows: where ρ is the fluid density; u i and u j are the mean velocity components corresponding to i and j, respectively; p is the pressure; μ is the fluid viscosity; and t is the time. e Reynolds stress tensor, τ ij � − ρu i ′ u j ′ , needs to be numerically modeled to close the equations.
An exhaustive investigation on a plane wall jet was conducted by Yan et al. [34] using seven Reynolds-averaged Navier-Stokes (RANS) turbulence models and a large eddy simulation (LES). ey found that the stress-omega Reynolds stress model (SWRSM) with adjusted turbulence model constants achieved the best results in simulating a steady wall jet without co-flow. In addition, Sengupta and Sarkar [35] indicated that the LES model, realizable k-ε model, and Reynolds stress model (RSM) perform better in simulating an impinging jet. In the current study, the Navier-Stokes equations were closed by employing RSM (stress-omega). e stress-omega model is a stress-transport model proposed by Wilcox [36], and a revised version was introduced subsequently [37]. e default constants from the original version [36] are used in FLUENT 16.0. e revised version was used in the current study. e parameters of the SWRSM are listed in Table 1.

Assumptions and Numerical Solution.
e assumptions made for solving the pressure and flow fields inside the wind channel and their corresponding implications are as follows: (1) Constant and uniform properties, i.e., ρ and μ are constant. e differential equations governing the flow were integrated using the finite-volume method, which is a specific case of residual weighting methods [38,39]. e least squares cell-based method was adopted for the numerical approximation of gradients, and bounded central differencing was selected for the momentum discretization. e pressure was discretized using a second-order implicit scheme. e discretized governing equations were solved using the Semi-Implicit Method for Pressure-Linked Equations Consistent (SIMPLEC) algorithm, which is an improved version of the SIMPLE algorithm [40], for pressure-velocity coupling. e SIMPLEC algorithm adopted in this study is presented in Appendix A. e time step was selected to guarantee that the Courant number was less than one to maintain the stability of the computation. e absolute convergence criteria for the continuity equation and the other equations are 1 × 10 − 6 and 1 × 10 − 5 , respectively.

Flow Configuration and Computational Setup.
e computational domains for the plane wall jet and impinging jet using the Cartesian coordinate system are shown in Figure 1, where b is the jet inlet height of the plane wall jet and D is the circular inlet diameter. For the plane wall jet domain, Yan et al. [41] indicated that the development of the wall jet is not affected up to the streamwise position e domain height of the plane wall jet assumed in this study was 21b. e bottom was set as a wall boundary with a no-slip condition. At the inflow plane, a velocity profile was set for the wall jet up to y/b � 1.0, and the rest of the plane had a uniform co-flow, U E . Jet entrainment was provided by a uniform co-flow. e top boundary was specified as a freeslip boundary condition. e spanwise direction had periodic boundaries to attain two dimensionalities, and at the exit plane, a pressure outlet boundary condition was applied, as shown in Figure 1(a). Sengupta and Sarkar [35] showed that the geometric conditions of the domain have little influence on the flow profiles. For the current impinging jet simulation, a three-dimensional (3D) cylindrical domain was used, as illustrated in Figure 1(b). Pressure outlet boundary conditions were applied with a zero-normal gradient at the outflow boundaries. A no-slip condition was assumed at the bottom wall of the computational domain. e turbulence intensity of the inflow was set as 0.01 for both the impinging jet and plane wall jet.
A nonuniform hexahedral grid was used. e nearest node to the wall in the y-direction was located at y + <1 for all grids, where y + � Δyρu τ /μ is the nondimensional wall distance, u τ � (τ w /ρ) 1/2 is the friction velocity, τ w is the wall shear stress, μ is the dynamic viscosity of the fluid, and ρ is the density of the fluid. To ensure grid independency, two grids were employed for the radial wall jet simulation: coarse G1 (2.4 million) and fine G2 (4.1 million). Two grids were also employed for the plane wall jet simulation: coarse G3 (1.8 million) and fine G4 (3.2 million). As shown in Figure 2, the results for the two levels of grid resolution were very similar, and there was no noticeable difference between the mean velocities of the radial and plane wall jets. All the results presented in this work are for the two fine grids shown in Figure 1. e mean velocity is usually scaled with length and velocity variables, namely, the maximum velocity U m and the distance y 1/2 from the wall to the position at which the mean velocity declines to half of its maximum value [7,42], to obtain self-similar characteristics. Researchers often selected the height y m of the maximum velocity location as the length scale [43][44][45] for the vertical mean velocity profile of the downburst.
Hjelmfelt [46] noted that approximately 50% of the observations from the Joint Airport Weather Studies (JAWS) Project were for traveling events. Storm translational velocities can be as high as 1/3 of the downdraft velocity. A downburst with "surface environmental wind" or a downburst embedded in a translating storm can be modeled through a plane wall jet approach with a co-flow [32,33]. e velocity ratios are defined as β � U E /U j for the plane wall jet. e Reynolds number is defined as Re � U jet D/ v for impinging jet and Re � U jet b/v for plane wall jet, where v is the kinematic viscosity. e simulation cases for investigating the influences of the Reynolds number and nozzle height on the impinging jet are listed in Table 2, while those for investigating the influences of the Reynolds number and co-flow on the plane wall jet are listed in Table 3.

Numerical Procedure Validation.
To verify the reliability and accuracy of the current simulation, the results obtained from the current model were compared with the experimental results obtained by McIntyre [47] for a Reynolds number of Re � 30,700 and a velocity ratio of β � 0.1. Because the main objective of this study was to characterize the length and velocity scales, normalized mean velocities were used for validation. Figure 3 shows the velocity profiles at two downstream locations. As demonstrated in this figure, there is a good agreement between the current numerical results and experimental data from McIntyre [47]. It was concluded that the present numerical method is valid and can be used to predict the mean flow properties of wall jets.

Mean Axial Velocity of Impinging Jet.
e axial velocity profile along the centerline of the jet at half of the nozzle height from the ground plane is shown in Figure 4. In order to obtain the nondimensional velocity profiles, the mean axial velocity of the impinging jet is normalized by the local maximum velocity V m , and the radial distance is normalized by the local jet half-width δ 1/2 (defined as the width at which the mean axial velocity has decreased to half of its maximum value). L is the downstream distance from the jet nozzle. It can be seen that the current results of impinging jet match well with the experiment of Sengupta and Sarkar [35] on the right side and there are some differences on the left side. is discrepancy may be due to the concentration of seeding particles along the jet boundary, which is a common problem in the use of PIV. Compared with the velocity profile near the nozzle of a round free jet [48], the profile of impinging jet has a top-hat shape due to the existence of the plate. After impacting the plate, the axial velocity of the impinging jet is transformed into the radial velocity, while the round free jet gradually develops and becomes self-similar [27].

Vertical Profile of Mean Streamwise Velocity in Wall Jet
Region. Figure 5 shows the profiles of the mean velocity normalized using U m and y 1/2 at x � 1.5D for the radial wall jet with Re � 50,000 and H � 2D and at x � 30b for the plane wall jet with Re � 60,000 and β � 0.1. e current numerical results were compared with those of the plane wall jet experiment by Eriksson et al. [17]; the radial wall jet experiment by Cooper et al. [13]; and three empirical models for the vertical profile of the downburst [7,44,49]. e current Wall Inlet results of the plane wall jet are in good agreement with the vertical profile suggested by Wood et al. [7] and the results obtained by Eriksson et al. [17] for the outer region (y > y m ).
However, the current model underestimated the height of maximum velocity (y m ) by 7.8 and 4.9% and overestimated the velocity by 9.1 and 5.2% at the height of y/y 1/2 � 0.08, compared with those obtained by Eriksson et al. [17] and Wood et al. [7], respectively. e current results of the radial wall jet exhibit a larger y m and agree well with the vertical profile suggested by Oseguera and Bowles [49] for the region of y/y 1/2 > y m . In the inner region, the current model predicted a velocity lower by 10.1 and 11.8% compared to the experimental results of Cooper et al. [13] and Wood's profile at the height of y/y 1/2 � 0.08, respectively. It can be concluded that both the approaches can generate a flow that is similar to a downburst outflow and are effective in investigating downburst outflows. Figure 6 shows the distribution of RMS (root mean square) fluctuations in streamwise velocity profiles at different streamwise locations (2 < x/b < 3.5) and radial locations (40 < r/D < 70). It can be seen that the results from both wall jet and impinging jet are in good agreement with the literature data in the outer layer. e streamwise RMS velocity profiles of the wall jet show obvious twin-peak behavior and self-similarity, while there is no obvious peak near the wall in the simulation and experimental data of the impinging jet. Advances in Civil Engineering Figure 7 presents the RMS profiles of vertical velocity at different streamwise locations (x/b) and radial locations (r/ D). e numerical results of the plane wall jet are in good agreement with the experimental data and show obvious self-similarity, while the comparison for impinging jets with the hot-wire data sets is less good. e numerical results of the impinging jet are close to the experimental data of Knowles and Myszko [14] but smaller than the experimental results of van Hout et al. [12]. e profiles of Re shear stress of both plane wall jet and impinging jet also have two peak values. Different from streamwise RMS velocity, the peak values of Re shear stress near the wall are negative. e results of the plane wall jet match well with the experimental data of Rostamy et al. [18] while the result from Eriksson et al. [17] is smaller than the current simulation. e    Advances in Civil Engineering impinging jets values measured by Knowles and Myszko [14] are also considerably lower.

Effect of Jet Reynolds Number.
e dependence of the flow on the Reynolds number was studied for a fixed velocity ratio (β � 0.1). e decay of the spread of the jet flow and the maximum velocity was investigated. e average spread rate of the plane wall jet was found to be linear and can be expressed as the half-height (y 1/2 ) with respect to the downstream distance [10]. e growth is represented by equation (2) as follows:

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Many previous studies have reported the values of the slope A p . Launder and Rodi [10] summarized a large number of experiments and found that most values of A p fall within the range of 0.073 ± 0.002, except those obtained in low-Reynolds-number tests. e experiments of Eriksson et al. [17], where measurements were performed using LDV, indicated that the spread rate should be 0.078 for Re � 10,000. e experimental data from Abrahamsson et al. [16] indicated a dependence of the slope on Re and reported that values of A p varied from 0.075 to 0.081 with Re � 10,000-20,000. Wygnanski et al. [19] also found a clear Reynolds-number dependence and a larger slope value of 0.088 for a Reynolds number ranging from 3,700 to 19,000. With an external stream, Zhou and Wygnanski [20] indicated that the influence of the Reynolds number is less significant when the velocity ratio is large. Figure 9 shows the streamwise growth of the jet halfheight for different Re values with a co-flow ratio of 0.1. e values of the slopes varied from 0.0781 to 0.0733 with Re � 20,000-100,000. e obtained results and the experimental data from Abrahamsson et al. [16] are in good agreement for Re � 20,000. e Reynolds number does not seem to have a significant effect on the spread rates in the current study. e decrease in the spread rate tended to gradually decrease as the Reynolds number increased. A similar result was also observed by Abrahamsson et al. [16]. Schwarz and Cosart [50] reported that the variation in the spread rate was not apparent in their study for higher Re numbers ranging from 13,510 to 41,600. erefore, there should exist a threshold, which was found to be 60,000 in the current study. When the Reynolds number is greater than this threshold, the dependence of the spread rate on the Reynolds number can be ignored. When the wall jet approach is used to simulate the downburst outflow in the boundary layer wind tunnel, the Reynolds number is usually greater than this threshold. e large-scale features of the simulated outflow were independent of the Reynolds number.
Few studies have been conducted on the maximum velocity height (y m ). Zhou and Wygnanski [20] reported that y m exhibits an approximately linear relationship with the downstream distance, and the Reynolds number has no significant effect on y m . However, Reynolds-number dependence is observed in the CFD results obtained by Ben et al. [51] who found that y m decreases linearly with x at a higher rate for a lower Reynolds number. Figure 10 shows the longitudinal distributions of y m for different Re numbers.
e results from the current study are in agreement with the observations by Zhou and Wygnanski [20]. e growth rate of the maximum velocity height remained constant at dy m /dx � 0.0133, whereas in the experiment by Zhou and Wygnanski [20], the value was dy m / dx � 0.0114.
In previous studies [11,19], the decay of the maximum velocity is represented by the following equation: where x 0 is the virtual origin of the wall jet. e virtual origin is used to make the lines fit to converge the data to U m /U j � 1 at x � x 0 [19]. Velocity decay is well documented for the plane wall jet; however, the values of the coefficients B p and  Advances in Civil Engineering [52], and Tang et al. [26], respectively. However, Wygnanski et al. [19] also suggested that their results fit the power law fairly well, with N p � 0.5. Lin [53] found that the arithmetic mean values of N p and B p in previous studies were − 0.52 and 4.19, respectively. Barenblatt et al. [54] indicated that N p � 0.5 is necessary for achieving a completely similar flow.
In the current study, the power law with N p � 0.5 was used to fit the results. us, equation (4) can be rewritten as: e effect of the Reynolds number on the decay of the maximum velocity (U j /U m ) 2 is shown in Figure 11. e values of B p decreased with increasing Reynolds number and varied from 0.081 to 0.072 with Re � 10,000-100,000. e variation was approximately 13%. When the Reynolds number was higher (Re > 20,000), the variation in the coefficient B p was insignificant. For example, the value of B p varied from 0.074 to 0.072 with Re � 60,000-100,000. e variation was only approximately 2%. is indicates that the effect of the Reynolds number is negligible for higher values (Re > 60,000). is is in complete agreement with the observations made by Schwarz and Cosart [50].

Effect of Velocity Ratio.
e effect of the velocity ratio on the streamwise development of the half-width and maximum mean velocity for a fixed inlet Reynolds number was investigated. e Reynolds number has no significant effect on the rate of spread and the decay of the maximum velocity for Re > 60,000, as explained in Section 3.3.1. us, the simulations of the plane wall jet with various velocity ratios were conducted for Re � 60,000. e effect of the velocity ratio on the spread rate at half-height is shown in Figure 12. e value of A p was between the measured values from Zhou and Wygnanski [20] and McIntyre et al. [55], at a velocity ratio of 0.1. It can be observed that the velocity ratios have a significant effect on the half-height, which is in agreement with the findings by Zhou and Wygnanski [20]. e influence on the half-height, reported in Zhou and Wygnanski [20], is larger, which may be due to the low Reynolds number. e value of A p decreased as the velocity ratio increased, and the intercepts from various co-flow ratios were nearly the same. However, the velocity ratios also had no significant effect on y m , and dy m /dx was still 0.0133, as shown in Figure 13.
Few studies have reported the values of B p for different velocity ratios. To compare the decay of the maximum velocity downstream, the studies of Wygnanski et al. [19] and McIntyre et al. [55] were considered. e effect of the velocity ratio on the streamwise decay of the maximum velocity is shown in Figure 14. e result from the current study at β � 0.1 is similar to that of Wygnanski et al. [19].
McIntyre et al. [55] reported the coefficient B p � 0.052, which is 30% smaller than that in the current study. It was observed

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that the values of B p decreased as the velocity ratio increased, that is, the decay of the maximum velocity slowed down. e values of A p and B p obtained from the current data for different velocity ratios are shown in Figure 15. It can be observed that the velocity ratio had a significant influence on the evolution of the plane wall jet. e values of coefficient A p increased linearly with the velocity ratio β, while the coefficient B p decreased exponentially with an increase in the co-flow ratio β.
According to the above analyses, the effect of a high Reynolds number can be neglected. us, only the effect of the velocity ratio needs to be considered when the plane wall jet method is used to simulate the downburst outflow. e spread of the plane wall jet with co-flow can be expressed as equation (5) and the decay of the maximum velocity with downstream distance for different velocity ratios can be expressed as equation (6).

Effect of Jet Reynolds Number.
is section explains the effects of the Reynolds number on the jet spread and velocity decay for the wall jet region of the impinging jet. According to Hjelmfelt [46] summary of JAWS results, the average downburst diameter is approximately 1.8 km and the average distance from the cloud base to the ground surface is 2.7 km. On average, the ratio of the cloud base height to the downburst diameter is approximately 1.5. us, a widely used distance of H � 2D from the jet nozzle to the bottom wall [56] was adopted in the current study.
For a fixed nozzle height above the plate board, H � 2D, the variation in the half-height with r positions for different Reynolds numbers is shown in Figure 16. It can be observed that the Reynolds number has no significant effect on the half-height, which can be considered independent of the Reynolds number. Sengupta and Sarkar [35] and Li et al. [56] proposed empirical expressions for the distribution of halfheight based on experimental and CFD data, respectively.  e current CFD results are very similar to the empirical curve obtained by Sengupta and Sarkar as well as the hotwire data from Copper et al. [13] in the region of r < 3.5D. e empirical curve obtained by Li et al. [56] and the experimental data reported by Knowles and Myszko [14] exhibit larger values than those in this study. By fitting with the CFD results, the spread of the jet flow can be expressed as Because of the high convective heat transfer from the wall near the stagnation point, most of the studies on radial wall jets are limited to the stagnation region. e wall jet region of the impinging jet has been less emphasized. Xu and Hangan [45] suggested that the impingement region extended from the free jet axis to the location of r/D � 1.4. Tummers et al. [57] reported that the minimum value of half-height for an impinging jet is located at r/D � 1.5. Cooper et al. [13] indicated that the radial wall jet grows linearly with distance r > 2D, and the nozzle height has little effect on the slope of the jet growth. e results from Knowles and Myszko [14] exhibited a linear growth for r > 2.5D. Figure 17 shows the plot of the half-height with r for different Reynolds numbers and r > 1.8D. e slope obtained from the current study was 0.098, which is equal to the value reported by van Hout et al. [12]. e effect of the velocity ratio on the radial evolution of the maximum velocity decay, together with the available experimental results, is presented in Figure 18. measured results from different studies do not agree well with each other. It can be observed that the values of U m /U j for different Reynolds numbers in the same radial position are the same. e maximum mean velocity scaled with the slot quantities is independent of the Reynolds number. is is in agreement with the observations made by Xu and Hangan [45]. Sengupta and Sarkar [35] proposed an empirical expression as follows: where a � 1.905, b � 1.858, and c � 1.949. However, to fit with the CFD results of the current simulation, the values of a, b, and c should be adjusted to 2.617, 2.637, and 2.27, respectively. For 1 < r/D < 2, the current results agree well with those of Knowles and Myszko [14] for Re � 90,000. For r/ D > 2, a good agreement can be observed between the current data and the results obtained by Xu and Hangan [45] for Re � 43,000. e maximum radial velocity (U m ) was almost equal to the jet velocity at the radial station for r/ D � 1.1. is is in agreement with the findings by Tummers et al. [57].

Effect of Nozzle Height.
To examine the effect of the nozzle height above the plate board on the evolution of the radial wall jet, the radial distributions of r location of the half-height for different nozzle height-to-plate distance ratios (H/D) are presented in Figure 19. e impingement region gradually decreased with an increase in nozzle height H. In general, the half-height value increases with the increase in nozzle height H, which is in good agreement with the results obtained by Knowles and Myszko [14] and Cooper et al. [13]. e current results are similar to those of Copper et al. [13] in the impingement region and are in good agreement with the data obtained by Knowles and Myszko [14] in the radial wall jet region. However, when the outflow height H/D < 2 or 3 < H/D < 5, the half-height did not change significantly. erefore, for the range in which these nozzle heights are located, the outflow height has little effect on the half-height value. e effect of the nozzle height on the radial evolution of the maximum velocity is shown in Figure 20. It can be observed that with an increase in the nozzle height, the closer the radial position of the maximum wind speed to the stagnation point, the lower the maximum wind speed. Compared with existing literature data, the current simulation results are similar to the data reported by Knowles and Myszko [14] in the impingement region, and they agree well with the results obtained by Cooper et al. [13] for r > 1.5D. e influence of the nozzle height on the maximum velocity decay also exhibited step characteristics, similar to the influence on the half-height.
Based on the data collected during the JAWS project, Hjelmfelt [46] found that the average ratio of the downburst outflow height to diameter is approximately 1.5. It can be observed from the above results that the nozzle height has no significant effect on the jet spread and velocity decay of the radial wall jet near this average ratio. erefore, when the  radial wall jet method is used to study the downburst, the influence of the outflow height can be ignored. e most widely used nozzle height is 2D [35,56].

Validation of the 2D Assumption for the
Downburst Outflow e outflow of the stationary ideal downburst radially spreads outward from the stationary point. When the radial wall jet and plane wall jet are used to simulate the outflow of the downburst, although similar wind profiles can be generated, it can be seen from the above studies that there are differences between the jet spreads and velocity decays. In comparison, radial wall jets have faster decay rates. To evaluate the accuracy of approximating a 3D downburst outflow with a 2D wall jet, the configuration of the transmission tower and downburst, as presented in Figure 21, was examined. A singlespan transmission tower-line system with a span length of S was used for the analysis. e distance r A and angle θ were used to define the location of the center of the tower relative to the stagnation of the downburst. e differences between the half-heights and maximum velocities of locations A and B are defined as Δy 1/2 and ΔU m , respectively. e deviation of the half-height differences of the downburst outflow obtained using the two methods is and the deviation of the maximum velocity differences of the downburst outflow obtained using the two methods is A downburst case was assumed with D � 1000 m and U j � 80 m/s to investigate the difference between the impinging jet and plane wall jet. Tower A is located at a distance of 1.8D from the stagnation point. Using simple trigonometry, the radial distance of tower B, which is the location of the stagnation point of the downburst relative to the center of the tower, can be evaluated. When the plane wall jet method is used, x a is equal to r a . Using simple trigonometry, the value of x b can be calculated. In addition, the height of the jet nozzle (b) can be estimated according to the height of the maximum wind velocity (y m ) and the diameter of the downburst (D). e detailed algorithm is provided in Appendix B. erefore, the length scale deviations between the radial and plane wall jet methods can be evaluated using equations (5), (7), and (9). e velocity scale deviations can be evaluated using equations (6), (8), and (10). For various combinations of angles (θ) and span lengths (S), the ratios of the length scale deviations to the half-height at location A for the radial wall jet are listed in Table 4, and the ratios of the velocity scale deviations to the maximum velocity at location A for the radial wall jet are listed in Table 5.
Assuming a ratio of less than 5% as an acceptable value, all the values in Table 4 and the upper left (unshaded) values in Table 5 indicate that the two-dimensional (2D) assumption is valid for wide structures. For example, for a 200 m transmission tower-line system, there are no clear differences between the length and velocity scales of the radial and plane wall jets. e simplified 2D approach appears to be effective for simulating downbursts. Using the plane wall jet method, a large-scale wind tunnel test can be performed based on the traditional atmospheric boundary layer wind tunnel.   Advances in Civil Engineering 13

Conclusions
In the present study, turbulent radial and plane wall jets were simulated numerically using RSM. e numerical results were compared with previous experimental measurements in the literature, and the effects of different parameters on the length and velocity scales were systematically evaluated. Based on the CFD results, it is valid to approximate a downburst outflow with a 2D assumption for a transmission line under specific conditions. e main findings of this study are summarized as follows: (1) e computed results show that the Reynolds stress model accurately predicts the behaviors of radial and plane wall jets. e predictions from the current simulation agree well with the experimental data available in the literature. Compared with the existing experimental results, the maximum difference was approximately 12%. Both radial and plane wall jet methods can effectively simulate the characteristics of the mean velocity profile of the downburst outflow.
(2) e decay of the maximum velocity and rate of jet spread for the radial wall jet are independent of the Reynolds number for a fixed nozzle height. e nozzle height has a clear effect on the evolution of the radial wall jet. However, when the value of H/D is approximately less than 2, which includes the average ratio of cloud base height to the diameter of most downbursts, the influence of the nozzle can be ignored.
(3) e decay of the maximum velocity and the halfheight of the plane wall jet are dependent on the Reynolds number below a critical value, Re cr � 60,000. Above Re cr , the flow becomes asymptotically independent of the Reynolds number. e influence of the Reynolds number can thus be neglected when the plane wall jet is used to simulate the downburst outflow for Re > Re cr . To improve the usability of the plane wall jet approach, the shape functions of scale parameters were proposed.
(4) Co-flow has a significant influence on the plane wall jet. With an increase in the velocity ratio, the jet spread and decay of the maximum velocity gradually slow down. When the velocity ratio increases from 0.1 to 0.35, the value of B P decreases by 52.7%. (5) Within the span length of a conventional transmission tower-line system, the discrepancy between the downburst outflows simulated using the plane and radial wall jet approaches in the longitudinal direction can be neglected. It is valid to approximate the downburst outflow with a 2D assumption from the perspective of the longitudinal evolution of the flows.
In large-scale facilities, the profile can be optimized by controlling the initial flow field conditions, and the plane wall jet method can produce a flow field several times larger than that of the radial wall jet method.

Appendix
A. The SIMPLEC algorithm e SIMPLEC algorithm uses the relationship between the velocity and pressure corrections to enforce mass conservation and obtain the pressure field.
To provide a brief review of the SIMPLEC method, the staggered grid shown in Figure 22 is used. e discretized umomentum equation can be written as follows: a e u e � a nb u nb + b e + A e p P − p E , (A.1) where p is the pressure, A e is the area of the face of the P-control volume at e, and a e is the coefficient of the finitevolume equations. A pressure field p * is assumed to initiate the SIMPLE calculation process. e u * velocity is obtained by solving the u-momentum equations and satisfy the following equation: a e u * e � a nb u * nb + b e + A e p * P − p * E .

(A.2)
However, the u * velocities from equation (A.2) in general do not satisfy the continuity condition. To correct the u * field, the estimated pressure is corrected by considering p′ � p − p * . Subtracting equation (A2) from equation (A1) gives a e u e ′ � a nb u nb ′ + A e p P ′ − p E ′ , (A.3) where p′ and u′ are the pressure and velocity corrections, respectively. e correct pressure p and velocity u can be written as For the control volume shown in Figure 22, the continuity equation satisfies (ρuA) w * − (ρuA) e +(ρvA) s − (ρuA) n � 0. (A.7) Substituting the correct u into the continuity equation gives the following result: e pressure correction p' can be obtained using equation (A8). Subsequently, the correct velocity field can be obtained.

B. Evaluation of Effective Downdraft Diameter for the Plane Wall Jet
Previous studies have shown that the maximum mean velocity occurs at a height of less than 0.05D [28,45,58]. For the radial wall jet, the value of y m obtained from the current RSM results at a distance of 1.5D from the stagnation point (x � 1.5D) is 0.03D. e influences of Reynolds numbers on the maximum velocity and location are negligible when Re > 60,000. An effective plane wall jet downdraft diameter can be determined based on the height of the maximum velocity, as expressed in equation (B1): For example, the nozzle height of the plane wall jet is 0.03 m in the current simulation; thus, the effective plane wall jet downdraft diameter is evaluated as 0.53 m at the downstream distance of x � 30b.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.