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Turbulence Transport Equations
Abstract
A generalized eddy viscosity function σ, is introduced in order that the Reynolds stress in an incompressible fluid be expressible as a linear combination of the Kronecker and rate‐of‐strain tensors. A transport equation for the eddy viscosity is derived from the general turbulence energy equation, thereby introducing two additional functions, the specific turbulence kinetic energy q, and a scale variable s. To determine the three variables, a transport equation for s is postulated, and a modified Prandtl—Wieghardt relation among the three variables is accepted. The theory is expressed in universal, invariant form, and validity is demonstrated by application to several problems.
... The k ε − model was demonstrated to provide reasonable approximations for various types of flows [31]. It consisted of two transport equations for the turbulent kinetic energy T k and its dissipation T ε [32]. ...
... The k − ε model was demonstrated to provide reasonable approximations for various types of flows [31]. It consisted of two transport equations for the turbulent kinetic energy k T and its dissipation ε T [32]. ...
Wave is a common environmental load that often causes serious damages to offshore structures. In addition, the stability for the submarine artificial slope is also affected by the wave loading. Although the landslide of submarine slopes induced by the waves received wide attention, the research on the influence of solitary wave is rare. In this study, a 2-D integrated numerical model was developed to investigate the stability of the foundation trench under the solitary wave loading. The Reynolds-averaged Stokes (RANS) equations were used to simulate the propagation of a solitary wave, while the current was realized by setting boundary inlet/outlet velocity. The pore pressure induced by the solitary wave was calculated by Darcy’s law, and the seabed was characterized by Mohr–Coulomb constitutive model. Firstly, the wave model was validated through the comparison between analytical solution and experimental data. The initial consolidation state of slope under hydrostatic pressure was achieved as the initial state. Then, the factor of stability (FOS) for the slope corresponding to different distances between wave crest and slope top was calculated with the strength reduction method. The minimum of FOS was defined as the stability index for the slope with specific slope ratio during the process of dynamic wave loading. The parametric study was conducted to examine the effects of soil strength parameters, slope ratio, and current direction. At last, the influence of upper slope ratio in a two-stage slope was also discussed.
... The k-ε model, which encompasses two transport equations for turbulent kinetic energy kT and its dissipation εΤ, represents a more intricate and widely utilized approach (Harlow, 1967). The k-ε model has demonstrated its ability to offer suitable approximations for various flow types (Rodi, 1980). ...
Scouring is one of the important issues that caused damage to the structures. Failure due to local scour has inspired many researchers to study the cause of scouring and to predict the maximum scouring depth around the bridge pier. Numerical simulation is proposed as an effective tool for monitoring the depth of scouring to manage the stability and safety of the bridge. Flow-3D is an accurate, fast, proven CFD software that can solve the toughest free-surface flow problems. However, the guideline information of this software is limited. Scouring classification and mechanism around bridge pier has been discussed briefly. The important things about the Flow-3D model setup are discussed. Verification by comparing the experimental and numerical results is required to determine the best model. Some studies of scouring simulation around bride pier by using Flow-3D software were presented in this paper to prove the accuracy of this software in predicting and simulating the scouring. Zhang's research study is selected as the best numerical model which has the closest result with the experimental result due to the error rate is 0%. This study used the Renormalized group (RNG) model as a turbulence model. For sediment scour model Soulsby-Whitehouse equation and Van Rijn equation are proved as the best model for Critical shields number definition and bed-load transport rate equation. The finer mesh size around the bridge pier was set up to get an accurate result. Specified velocity and outflow are used for the left and right boundaries. Moreover, for front and back boundary were using symmetry, and the bottom and top boundary were using the wall and specified pressure.
... The simulations were carried out using four different turbulence schemes: (i) the twoequation k − ω model [61][62][63]-suitable for modeling free shear flows with streamwise pressure gradients like spreading jets, wakes, and plumes; (ii) the two-equation k − ε model [64]-dynamically calculates the turbulent mixing length and thus useful for a wide range of flows [65]; (iii) the renormalized group (RNG) k − ε model [66,67]-extends the capabilities of the standard k − ε model and provide a better coverage for transitionally turbulent flows; (iv) the LES model-resolves most of the turbulent fluctuations directly, thus requiring much more computational resources compared to the two-equation models. Figure 7 highlights the implication of these four turbulence schemes in terms of scour and depositions in the vicinity of the structure for the side layout. ...
The present study evaluates the performance of two numerical approaches in estimating non-equilibrium scour patterns around a non-slender square structure subjected to a transient wave, by comparing numerical findings with experimental data. This study also investigates the impact of the structure’s positioning on bed evolution, analyzing configurations where the structure is either attached to the sidewall or positioned at the centerline of the wave flume. The first numerical method treats sediment particles as a distinct continuum phase, directly solving the continuity and momentum equations for both sediment and fluid phases. The second method estimates sediment transport using the quadratic law of bottom shear stress, yielding robust predictions of bed evolution through meticulous calibration and validation. The findings reveal that both methods underestimate vortex-induced near-bed vertical velocities. Deposits formed along vortex trajectories are overestimated by the first method, while the second method satisfactorily predicts the bed evolution beneath these paths. Scour holes caused by wave impingement tend to backfill as the flow intensity diminishes. The second method cannot sufficiently capture this backfilling, whereas the first method adequately reflects the phenomenon. Overall, this study highlights significant variations in the predictive capabilities of both methods in regard to the evolution of non-equilibrium scour at low Keulegan–Carpenter numbers.
... Trong đó: l và Mô hình chảy rối sử dụng hai phương trình k- [31] được ứng dụng rộng rãi trong tính toán động lực học dòng rối và tương tác dòng rối với kết cấu. Với các kết cấu công trình thuỷ lợi, đê chắn sóng [27], mô hình Renormalization Group (RNG) đã được kiểm chứng về hiệu quả mô phỏng tương tác dòng chảy -kết cấu [32], [33] và được sử dụng trong nghiên cứu này. ...
Dưới ảnh hưởng ngày càng lớn của biến đổi khí hậu, hàng loạt các bờ biển, bãi biển, hải đảo bị xói lở nghiêm trọng. Đê chắn sóng rỗng dạng ngập được sử dụng để giảm sóng tới một cách chủ động từ xa mà vẫn đảm bảo mỹ quan, không che chắn đối với các hoạt động tắm biển, du lịch. Tuy nhiên, do kết cấu đê rỗng và đỉnh ngập dưới mực nước tĩnh nên hiệu quả giảm sóng là không cao. Nghiên cứu này đề xuất biện pháp tăng bề rộng đê và chỉ dẫn cách thực hành tính toán xác định bề rộng đê tường đứng rỗng dạng ngập đáp ứng hiệu quả chiết giảm sóng đơn tới. Mô hình số Flow-3D được sử dụng để mô phỏng và xác định hệ số truyền sóng với 48 kịch bản khác nhau về chiều cao đê, bề rộng đê và chiều cao sóng đơn tới. Kết quả cho thấy sự phù hợp về hệ số truyền sóng đối với các trường hợp chiều cao đê và bề rộng đê ngập tăng dần. Các điểm quan hệ bề rộng đê - hệ số truyền sóng được sử dụng để thiết lập các đường hồi quy bậc hai. Hệ số truyền sóng và hệ thống đồ thị các đường hồi quy bậc hai được sử dụng để chỉ dẫn thực hành tính toán bề rộng đê tường đứng rỗng dạng ngập đáp ứng hiệu quả chiết giảm sóng đơn
... This also allowed more parameters to be accurately predicted, such as heat transfer and interactions between fluids and solids. A key step in CFD development came with the invention of the k-ε turbulence model (Harlow and Nakayama, 1967); these equations were eventually standardised by Launder and Spalding (1974) and have been widely used since. ...
У статті розглядаються проблеми впливу раціональних режимів роботи хлібопекарських печей, що є важливою задачею роботи печі для значної економії палива. У хлібопекарських печах однією з основних величин, найбільш чуттєвою до зміни навантаження, є температ
... Among the best-known turbulence models, there is the k-ε model [29], which is defined by a pair of transport equations for the turbulent kinetic energy k, and its dissipation ε. Over the years, this model has made it possible to approximate significant types of flows, although it is necessary to modify the dimensionless parameters. ...
Any scalar concentration discharging to the sea through a freshwater release is a thermodynamic two-phase flow including different densities and different temperatures. Because of Reynolds similarity, physical models are not feasible to use and numerical models such as CFD-RANS are the solutions. A series of numerical experiments have been conducted to derive an empirical formula that determines the length of the 95% scalar concentration reduction, useful for primary and emergency engineering-environmental decisions. Results show that the scalar concentration intrusion length is a function of seawater density-temperature, freshwater density-temperature, sea current Froude number, effluent jet Froude number, and the scalar concentration in which among those, the sea current Froude number has the most effective role in which as an example of the lowest and highest tested sea current velocities, an effluent discharge of 0.052 m 3 /s with a concentration of 1 Kg/m 3 and effluent jet velocity 0.245 m/s in 3.2 meters from the effluent source reaches to the 95% scalar concentration reduction when there is a high-velocity sea current (0.5 m/s) while in the presence of a low-velocity sea current (0.05 m/s), the same effluent discharge with the same scalar concentration and the same effluent jet velocity, the length of the 95% scalar concentration reduction increase to 42.8 meters from the effluent source.
... là ứng suất biên theo các trục tọa độ, các thành phần là ứng suất cắt. Mô hình chảy rối sử dụng hệ số nhớt động lực và hai phương trình tổn thất năng lượng k − , được thể hiện như sau [17]: ...
Tính toán mô phỏng dòng chảy sau vỡ đập được thực hiện phổ biến trên thế giới, nhất là trong hoàn cảnh biến đổi khí hậu đang diễn ra ngày càng phức tạp và tác động khó lường tới tính mạng con người cũng như cơ sở hạ tầng. Bài báo trình bày tương tác dòng chảy sau vỡ đập với các trường hợp đáy nhám phức tạp sử dụng mô hình dòng chảy ba chiều Flow 3D. Mô hình giải tích và mô hình thí nghiệm vật lý được tiến hành cho trường hợp đáy phẳng không nhám để kiểm chuẩn kết quả từ mô hình Flow 3D. Hai trường hợp đáy nhám phức tạp, tương ứng là đáy nhám có khối tam giác và đáy nhám có khối hộp chữ nhật, được lựa chọn để thể hiện tương tác với dòng chảy sau vỡ đập. Với trường hợp đáy nhám có khối tam giác, mô hình số đã mô phỏng chính xác đặc trưng dòng chảy qua đáy nhám và chiều cao dòng chảy tràn qua đáy có khối tam giác. Với trường hợp đáy nhám có khối hộp chữ nhật, vận tốc dòng chảy trước và sau tương tác với khối hộp chữ nhật được mô phỏng khá phù hợp với kết quả thí nghiệm mô hình vật lý.
... The CFD model uses the Navier-Stokes equations of incompressible fluid and the k-ε turbulence model [31]. The governing equations of incompressible fluids are solved using the finite volume method. ...
Marine submersible buoy systems hold significant value as critical equipment in marine science research. This study examines a marine submersible buoy system that includes an anchor block, mooring line, battery compartment, power supply cable, and submersible buoy. The anchor-last deployment method is a conventional strategy for deploying marine submersible systems. Initially, the other components are positioned on the sea surface, followed by the deployment of the anchor block from the ship’s deck. The anchor block will pull the battery compartment and submersible buoy into the water and eventually sink to the seabed. In this deployment process, ocean currents have a relatively large impact on the anchor block’s landing position. Increasing the weight of the anchor block will make the anchor block land on the seabed sooner, which can minimize the impact of ocean currents. However, an overabundance of weight can generate a significant strain on both the cables, potentially resulting in cable breakage. In order to find the parameters that can make the anchor block reach the seabed as soon as possible and ensure that the tension force of the cables does not exceed the maximum, a dynamic model of the deployment process is established based on computational fluid dynamics (CFD) and solved using the Runge–Kutta method of the fourth order. Particle swarm optimization is employed to optimize the key parameters. The penalty function is used to constrain the particle space. The findings indicate that the utilization of particle swarm optimization is efficacious for optimizing the parameters of submersible buoy systems for marine applications. Optimized parameters allow the anchor block to reach the seafloor quickly and the tension on the cables to not exceed the given value.
... The standard k-ε model (Harlow and Nakayama 1967) is a two-equation model that dynamically determines the turbulent mixing length TLEN while calculating the turbulent kinetic energy (k) and dissipation rate (ε). It serves as an industry standard and may be used to represent a variety of flows (Turbulence Model 2022; Rodi 1980). ...
The computational fluid dynamics (CFD) method is used effectively in hydraulic engineering and many other sciences. However, determining which turbulence model is suitable for the analysis requires further investigation. This study aims to show which turbulence method is closer to the actual data in calculating parameters, such as velocity, water surface profile, and pressure, frequently encountered in CFD engineering. For this purpose, the discharge-water level, pressure, energy dissipation rate, and velocity profile were investigated using different turbulence models (k–ε, k–ω, large eddy simulation [LES], renormalization group [RNG]). Then the results were compared with the physical results of stepped spillways. According to the results, the most compatible turbulence model in the discharge-water level relationship is k–ω; the most compatible turbulence model is k–ε for pressure, energy dissipation rates, and approach channel velocities; and lastly, the most compatible turbulence model was LES for water surface profiles. The results obtained are expected to be a reference for researchers who will work in this field.
... In the past [49], k-epsilon method has been reported as a suitable turbulence model to simulate confined and free-surface flows, which is the case of realistic footwear slipping on flooring surfaces. Equation 4 and Eq. 5 represent the k-epsilon turbulence model [50], where, P k is the turbulence production due to viscous forces, t is the turbulence viscosity, and C , C 1 , k , , and C 2 are constants. These constants were applied with values 0.0845, 1.42, and 1.68 respectively [46]. ...
Slips and falls are prevalent across the globe and pose major threat to the safety of public. Significant frictional performance of a footwear is essential to mitigate the traumatic injuries caused due to unintentional slips and falls. Several studies in the past have evaluated the traction performance of the footwear across common slippery conditions, but only a few have considered testing the worn shoes. In this study, nine systematically altered tread designs were implemented to study the effects of tread geometries across its unworn and worn conditions. The outsoles were experimentally tested for their friction performance and also simulated for wet slippery conditions using a novel computational fluid dynamics based framework. Outsoles with a tread width of 6 mm and tread gaps of 2 and 3 mm showed higher friction coefficients over dry surfaces. After first wear phase, outsoles with 2 and 3 mm gaps exhibited higher friction outcomes as compared to higher tread gaps (i.e., 4 mm). Outsoles with tread width of 2 mm and tread gaps of 2 and 3 mm showed retention of treads, to some extent, and also exhibited the highest friction values even after third wear phase. Considering all the parameters (i.e., tread gap, tread width, friction values and wear phases), outsole with tread width and gap of 4 mm exhibited considerably higher friction values. The results from this study are anticipated to provide better understanding of the effect of footwear tread parameters on their frictional performance. Furthermore, the procedures used in this study would also help the footwear manufacturers to determine the footwear replacement thresholds to mitigate the slip and fall risks.
... In Flow-3D, three types of turbulence models are built into the solution: the standard k-ε model (Harlow and Nakayama, 1967), the renormalization group (RNG) model (Yakhot and Orszag, 1986), which is an appropriate model for estimating the transport coefficients in the advection of a passive scalar by incompressible turbulence and the k-ω model (Wilcox, 2006) which is the most suitable for modeling free surface flows with streamwise pressure gradients like spreading jets, wakes, and plumes that is confirmed by Vanneste and Troch (2015) and Devolder et al. (2017) as well. Therefore, the k-ω model has been used as the turbulence model in the current study. ...
A new formula is introduced to determine the wave transmission at submerged porous breakwaters highlighting the effect of the wave-induced pore pressure distribution inside the breakwater, applicable for both regular and random waves. A CFD numerical wave flume based on the Stokes-Cnoidal wave theory for regular waves and Pierson-Moskowitz spectrum theory for random waves combined with the Forchheimer formula is calibrated using an experimental dataset from Calabrese et al. (2002). The proposed formula for the wave transmission coefficient has been obtained by means of dimensional analysis and incomplete self-similarity, and it has been calibrated using the results of the numerical experiments. The new formula has been verified using a large database on wave transmission coefficient yield fairing good predictions for a wide range of wave conditions and breakwater geometries.
The proposed formula has been compared with other existing formulae applying to 2336 data of the existing wave transmission coefficient dataset where the proposed formula giving the minimum root mean square error Erms = 12.6% showed the maximum accuracy among all other existing formulae for determining the wave transmission coefficient over the submerged porous breakwaters.
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... Two equation k-ε model computes turbulent kinetic energy and its dissipation rate. In this way, it finds mixing length dynamically [53]. The most robust version of k-ε model is RNG model because it explicitly calculates the constant coefficient of k-ε model [54]. ...
Trash racks are usually composed of an array of bars installed in a hydropower scheme to safeguard the turbines by collecting water-borne detritus. However, current design approaches for the design of trash racks focus on structural criteria. A little attention renders the proper evaluation of hydraulic criteria, which causes a significant hydraulic head loss in low head hydropower schemes with an integral intake. This study investigates the head loss through trash racks by employing computational fluid dynamics (CFD) for several design combinations. A three-dimensional model of trash racks using fractional area/volume obstacle representation (FAVOR) method in FLOW-3D is set up to define the effects of the meshing on the geometry and several simulations are carried out considering various approach velocities and different bar spacings, inclination angles, and blockage ratios. The results indicate that head loss increases with an increase in approach velocity, the inclination angle of the rack with channel bed, and blockage ratio. It is noticed that a clear spacing between vertical bars greater than or equal to 0.075 m has a minimum head loss before it becomes significantly high for lower spacing. In addition, the head loss coefficient increases for screen angles greater than 60°, which can be considered as an optimal parameter for design purpose.
... Harlow and his colleagues also contributed to the early numerical modeling of turbulence and, in particular, by the postulation of the now ubiquitous k-ε model in the 60's [51] [52]. The history of the early turbulence modeling in T-3 is included with the current modeling in section 3.5 below, in order to present a unified treatment of this complex topic. ...
The early history is presented of the prolific development of CFD methods in the Fluid Dynamics Group (T-3) at Los Alamos National Laboratory in the years from 1958 to the late 1960's. Many of the currently used numerical methods –PIC, MAC, vorticity-stream-function, ICE, ALE methods and the k- ε method for turbulence– originated at Los Alamos during this time. The rest of the paper summarizes the current research in T-3 for CFD, turbulence and solids modeling. The research areas include reactive flows, multimaterial flows, multiphase flows and flows with spatial discontinuities. Also summarized are modern particle methods and techniques developed for large scale computing on massively parallel computing platforms and distributed processors.
... 16,19 Spalding, Harlow, Kolmogorov, and Zhou Peiyuan have constructed "turbulence model mode", created many models, and developed some characteristic numerical calculation methods and programs. [20][21][22] After that, scholars introduced flow and combustion into the combustion link of heating furnace based on this model and achieved good results in practice. [23][24][25] The efficiency of steel heating furnace was improved by Han et al. 23 using the numerical simulation method, when oxygen fuel combustion was used instead of air-fuel combustion. ...
Heating furnace is an important energy‐consuming equipment in oilfield. In order to evaluate the operation status of heating furnace more accurately, in this paper, the production process of heating furnace in an oilfield is studied by numerical simulation technology, and the specific reasons affecting its thermal efficiency are deeply explored. First, the numerical simulation model of oilfield heating furnace production process is established, which is to digitize the geometric structure of the heating furnace, study the characteristics of the geometric shape, and generate the appropriate calculation grid. Second, numerical simulation and analysis are carried out for the working process of the heating furnace under rated and nonrated conditions, respectively. The medium flow characteristics, flame shape, temperature distribution, and pipe temperature distribution in the heating furnace under different conditions are systematically discussed. Finally, through the simulation analysis, it is found that the burner rotating device is beneficial to strengthen the heat transfer, and the change of load will affect the efficiency of the heating furnace. Under the rated working condition, design exhaust temperature is 260℃, numerical result is 225.8℃, and the error is −13.2%; design export temperature of water is 60℃, numerical result is 58.1℃, and the error is −3.2%; design temperature difference between inlet and outlet of water is 30℃, numerical result is 28.1℃, and the error is −6.3%. The working process of oilfield heating furnace is a complex coupling heat transfer process of combustion, flow, and heat transfer. The numerical results show that the working process of oilfield heating furnace could be successfully realized by means of numerical simulation. Under variable load conditions, in order to ensure that the export temperature of water is close to design temperature, it is necessary to change the fuel quantity in equal proportion to the water quantity. It is hoped that the reported results provide interesting information among the oilfield heating furnace operators.
... According to the three performance assessment indices, the k-ɛ model performed better, which also has been used in many previous dam-break studies for rapidly varied unsteady flows of high Reynolds numbers [17,[31][32][33][34][35]. k-ɛ [36] is a semi-empirical model which calculates k and ɛ (turbulence kinetic energy and dissipation rate) [34] and finds LT (i.e., turbulence mixing length). In addition, the performance of the three turbulence models to simulate water level is shown in Fig. 5. Based on Fig. 5, all three models succeeded in simulating flow condition, but k-ɛ model was more successful to simulate peak water level at approximately τ = 7 (τ = non-dimensionalised time). ...
Dam-break flow is known as one of the most horrible phenomena. Some hypothetical reservoir geometries were evaluated in literature, but in nature, each reservoir has a unique geometry. In the present research, dam-break flow was studied based on different reservoir geometries using FLOW-3D. Six reservoirs were considered: reservoirs R1 and R2 belonged to Mahabad Dam (Iran) and Tignes Dam (France), with asymmetric reservoirs, respectively; reservoirs R3 and R4 had symmetrical trapezoidal reservoirs with angles 30 and 45 degrees, respectively; reservoir R5 had a rectangular shape, extending from one side; and reservoir R6 had a long reservoir, which also was used to verify FLOW-3D. The model performance was verified by experimental results and FLUENT model in literature. Results showed FLOW-3D with mesh sizes 30×30×30 mm and k-ɛ turbulence model outperformed FLUENT, based on R2, RMSE, and MAE. The results of water levels and flow velocities at five points proved that dam-break flow could vary from one dam to another, considering reservoir geometry. Peak water levels and velocities have been measured to show how reservoir geometry could cause catastrophic flow.
... The simulation assumed a steady state condition and working fluid air K Epsilon turbulence model, continuity and momentum equations were used to obtain air mass behavior, pressure and velocity, the values of inlet wind velocity is 6 m/s and outlet set as atmospheric pressure, Domain walls, roof and floor are assumed smooth. [7], the flow was assumed fully turbulent, and no effects of molecular viscosity [8]. The standard k -Epsilon model is for fully turbulent air flow [9]. ...
These paper concerns the role of lattice windows in enhancing resident"s expectation of comfort ventilation in such places, especially unconditioned buildings which higher air speed is an important strategy to minimize discomfort from high temperature and humidity, especially that some architectural requirements like privacy call for placing openings in certain places that could lead to a poor ventilation in this occupied zone, unless air flow is directed by using some architecture elements in a desired direction by using special inlet opening like lattice windows. A model room used in Computational fluid dynamics CFD simulation program Ansys Fluent version,14.5 to figure out how lattice window delivers nature ventilation, Windows geometry, wing walls and side windows were observed how they affect wind direction and pressure difference to figure out its impact on room ventilation, A field measurement study in multi-story building with lattice windows were made by using Nova-Lynx weather station to observe internal ventilation speed and distribution with synchronous external wind.
... The standard k−ɛ model was considered as the turbulence closure for the Reynolds-averaged Navier-Stokes (RANS) equations to account for the turbulent wave-girder interaction. The standard k−ɛ model by Harlow and Nakayama (1967) is widely used to simulate dam-breaking waves, for example, Shigematsu et al. (2004), and the renormalization group (RNG) k−ɛ model by Yakhot and Smith (1992) is usually used in the simulation of the dam-breaking waves, for example, Robb and Vasquez (2015). Yang et al. (2019) conducted a comparative study of RANS equations with different turbulence models (standard k−ɛ, RNG k−ɛ, and k−ω) to show which turbulence model could accurately simulate dam-breaking wave generation and propagation. ...
This paper will present a comparative study of tsunami-induced forces on box girder and T-girder bridges, which will be based on component level analysis. Component level analysis can explain the overall loading behavior and can further illustrate the reason for the difference in the forces between these two types of girder. First, in the numerical simulation, a three-dimensional (3D) dam-breaking model (at 1/20-scale) will be developed to generate bore-type tsunami waves. The Reynolds-averaged Navier–Stokes (RANS) equations, combined with the k−ɛ turbulence model will be utilized for the wave simulations. Then, the effectiveness of the numerical model will be verified with the experimental results. The tsunami-induced forces on the box girder and T-girder bridges will be compared and the differences will be discussed in detail based on the component level analysis. In addition, parametric analyses will be conducted to study the influence of the wave momentum flux and still water level (SWL). The results show that: (1) the T-girder bridge witnesses higher and longer-lasting horizontal peak forces. The box girder bridge had significantly larger upward forces than the T-girder bridge, (2) for both type of girders, the upstream web and upstream deck were the major contributors to the maximum horizontal and vertical forces, respectively. Special attention should be paid to the local damage to these components, (3) when the wave is high enough to impact on the whole girder, the differences caused by the girder shape on the horizontal impulse loads can be negligible, and (4) the difference in the vertical impulse loads between these two types of girders continually increases with the momentum flux. For the large momentum flux cases, the vertical impulse loads on the box girder could be 1.7–2.2 times that on the T-type girder.
... The k-ε model is highly sophisticated, widely-used and consisted of two transport equations: one for turbulent kinetic energy k, and the other for its dissipation ε [20]. The 3D governing equations could be expressed as follows [21,22]: ...
Channel confluences are of the common structures in fluid transport channels. In this study, a series of numerical simulations were performed, utilizing a 3D code to investigate the reaction of the flow parameters and vortical structure to the variations in flow discharge and its Froude number from both main channel and tributary branch in a T-shape junction. The code was calibrated with the experimental data. Parameters, including the velocity, the turbulence energy, stream surface profile, head losses, and the transverse flow motions, were considered in different situations. It was concluded that increasing the ratio of discharge of flow from side-channel to the main channel (Q*) increased the area and power of the recirculation zone, as well as the width of separation plate downstream of the confluence, while it reduced the area of the stagnation zone (or the wake vortex) within the side-channel. It was also indicated that increasing the discharge ratio from side-channel resulted in an increase in the upstream water level in the main channels, which was dependent on the upstream discharge.
... While the work at Imperial College may have had the greatest early impact on turbulence modelling, there were several groups in the USA that had taken up the task of turbulence model development (or would shortly do so) though sometimes without rigorous testing. Harlow and Nakayama [28] announced: 'Our goal has been to develop transport equations that enable the macroscopic manifestations of turbulence to be included in high-speed computer calculations … without introducing such complexity as to render the equations intractable'. They chose a two-equation eddy-viscosity model in which the scale equation was proportional to the Taylor microscale. ...
This contribution presents the authors’ view of the historical evolution of modelling turbulence by way of the simple (though, some would say, outrageously simplistic) notion that the local turbulent stress–strain connection should be the same as in a laminar Newtonian flow. The principal emphasis is on modelling at a level where two transport equations are solved for scalar properties of turbulence, the level of approximation popularized (though not invented) by D. B. Spalding at Imperial College in the early 1970s. The successes and failures of the approach are examined. The chapter concludes by showing examples of closure at eddy viscosity level of what would be regarded as steady flows though treated by way of a time-dependent solution of the transport equations. These lead, in appropriate circumstances, to time-dependent structures which contribute additional momentum and heat transport thereby enhancing agreement with experiment.
... Among the two-equation models, three of the most extended options were tested for the present study. Firstly, the k-ε model [36,37] was tested. This model involves two transport equations, one for the turbulent kinetic energy (k) and another one for its dissipation rate (ε). ...
Adaptation of stilling basins to higher discharges than those considered for their design implies deep knowledge of the flow developed in these structures. To this end, the hydraulic jump occurring in a typified United States Bureau of Reclamation Type II (USBR II) stilling basin was analyzed using a numerical and experimental modeling approach. A reduced-scale physical model to conduct an experimental campaign was built and a numerical computational fluid dynamics (CFD) model was prepared to carry out the corresponding simulations. Both models were able to successfully reproduce the case study in terms of hydraulic jump shape, velocity profiles, and pressure distributions. The analysis revealed not only similarities to the flow in classical hydraulic jumps but also the influence of the energy dissipation devices existing in the stilling basin, all in good agreement with bibliographical information, despite some slight differences. Furthermore, the void fraction distribution was analyzed, showing satisfactory performance of the physical model, although the numerical approach presented some limitations to adequately represent the flow aeration mechanisms, which are discussed herein. Overall, the presented modeling approach can be considered as a useful tool to address the analysis of free surface flows occurring in stilling basins.
... The RNGbased models rely less on empirical constants, while setting a framework for the derivation of a range of models at different scales. The RNG model uses equations similar to the equations for the k-ε turbulence model (Harlow and Nakayama, 1967), but the equation constants that are found empirically in the standard k-ε model are derived explicitly in the RNG model. Therefore, the RNG model has wider applicability than the standard k-ε model. ...
Turbidity currents are a variety of subaqueous sediment-gravity flows, in which the suspension of sediment by water turbulence produces a water-sediment mixture that is denser than the ambient water and hence flows due to gravity along a topographic gradient. This type of sediment gravity flow is the most important mechanism for the dispersal and deposition of sand on deep-sea floors, as well as on the underwater slopes of many deltas and lakes.
The hydrodynamics of turbidity currents are difficult to study in the natural environments, whereas laboratory experiments are limited to small-scale flows, time-consuming and not necessarily easier when it comes to the measuring of flow properties and establishing of the relationships between the turbulent flow structure and the transport and deposition of sediment. Mathematical models of turbidity current, integrated by computational fluid dynamics (CFD) and realized as numerical simulations, can be used to obviate these difficulties, and also to upscale laboratory datasets and to integrate the data
from nature and experiments. The concept CFD refers to the numerical solution, by computational methods, of the governing equations describing fluid flow: the set of Navier-Stokes equations and the multi-phase fluid dynamics. CFD is widely used in the engineering branches of fluid mechanics, but is a relatively new numerical approach in the field of sedimentological research.
In the present study, a three-dimensional model has been constructed by using the CFD software Flow-3D™ to simulate the flow of turbidity currents, including their internal hydraulic characteristics as well as sediment erosion and deposition. The Flow-3D™ model employs finite difference and finite volume methods and the turbidity current is being simulated by a range of physical models: 1) the turbulent flow structure is simulated by a turbulence model based on renormalization group theory that
employs statistical methods to calculate turbulence quantities; 2) the water-sediment mixture is calculated by a drift-flux technique that describes the relative flow of two miscible fluids with different densities; 3) interactions between the continuous fluid and the dispersed mass particles are calculated by the particle model; and 4) the erosion and deposition of sediment are calculated by the sediment scour model.
Simulations of small-scale turbidity currents imitating particular laboratory flows have shown that the results of Flow-3D™ are realistic and reliable. Similar series of flow simulations have been used further: 1) to display in a flow-parallel axial section the main hydraulic characteristics (bulk density, shear-strain rate, dynamic viscosity, velocity magnitude and its x-y-z components) of a channel confined current; 2) to display of the shear stress, sediment concentration and velocity magnitude for several ‘probing stations’ in a channel-confined current expanding abruptly on an open-space flat floor; 3) to study the responses of the confined and unconfined parts of a current (in terms of its velocity
magnitude, shear stress and sediment concentration) to changes in such principal controlling parameters as the channel slope angle, sediment grain size, floor roughness and initial sediment concentration; 4) to display sediment grain-size segregation in a current (using a flow run with polysized sediment suspension); 5) to display the velocity time series for a surge-type and a sustained turbidity current; and 6) to show the responses of turbidity current to various obstacles and to a hydraulic jump at the channel outlet.
Two large-scale simulations of ‘real-life’ turbidity currents have been performed, one imitating modern flow events in the Soquel and Monterey canyons, offshore California, and another pertaining to the deposition of the Egga reservoir unit in the Ormen Lange field, Mid-Norway Continental Shelf.
The present study has evaluated Flow-3D™ as a possible means of simulating hydrodynamic behaviour of turbidity currents. The comparison of numerical and flume data indicates that the CFD based Flow-3D™ models can give realistic results and serve as an attractive alternative to laboratory flume experiments. The use of a CFD software, such as the Flow-3D™, has several great advantages: 1) it allows a much wider range of flow parameters to be determined and continuously monitored with a relatively high accuracy; 2) it permits the response and relative importance of the individual flow
parameters to be assessed with respect to changes in the initial conditions; 3) it allows turbidity currents to be up-scaled to natural conditions; and 4) it provides an unprecedented insight in the detailed hydrodynamic aspects of turbidity current. The study indicates that our understanding of turbidites and their variability can be significantly improved by this type of experimental research.
... The k-ε model [26] is calculated by two transport equations (refer Eq. 5) for the turbulent kinetic energy (k T ) and dissipation rate (ε T ). The equation for Diff ε is illustrated in Eq. (6) and kinematic turbulent viscosity ( T ) in all turbulence transport models is computed from Eq. (7). ...
Combined predictive modelling approach that couples the 2D far-field model and the 3D near-field computational fluid dynamic model has been developed to compare the performance of different turbulence models in predicting the flow kinematics around an open offshore intake structure. The k–epsilon (k–ε), k–ε renormalized group (RNG), k–omega (k–ω) and large eddy simulation (LES) models are used in this study. The objective is to select an appropriate turbulence model for simulating the flow field around the open offshore intake. The simulated water level and current speed of these turbulence models were compared with the field sampling results obtained at the vicinity of an existing open offshore intake structure located in the Penang Strait of Malaysia. Through the analysis, it can be concluded that the simulated water level for all turbulence models are broadly consistent with the major trend of measured values with k–ε model reporting the best performance. Comparison of the simulated current speed with the field measurements show that the k–ε RNG model fits better than the other models. The analysis reveals that the LES model has slightly lower accuracy in predicting current speed around the existing intake structure.
... Due to the complexity of turbulence, we need to select the appropriate turbulence model for the simulation. The k-ε turbulence model proposed by Harlow and Nakayama [4] is by far the most widely used turbulent viscosity model. We adopted the standard k -ε double equation model with parameters of defaulted values in FLUENT to model the airflow of the inlet duct. ...
This paper simulates the flow and the pressure distribution of the gas in the inlet duct in front of the draught fan in service for the power plant, by using the computational fluid dynamics method. Based on the prototype of the inlet duct, two feasible modifications of the structure were proposed to optimize the airflow distribution and to reduce the air pressure drop. Comparison of their numerical results indicated that the modification with a rounded elbow and inner air deflectors is optimal to reduce the airflow resistance of the inlet duct.
... Numerical modeling is carried out by means of a computational fluid dynamics model (CFD). Two different turbulence closure schemes are used, namely the standard k − ε turbulence model [32] and the LES model [33]. A volume of fluid (VOF) model is used to account for free surface effects [34]. ...
Gravity currents generated by lock release are studied in the case of initially quiescent ambient fluid and oscillating ambient fluid (regular surface waves). In particular, the dynamics of the density currents are investigated by means of CFD numerical simulations. The aim is to evaluate the influence of the ambient fluid velocity field on the observed mixing and turbulent processes. Results of two different turbulence closure models, namely the standard k − ε turbulence model and the LES model, are analyzed. Model predictions are validated through comparison with laboratory measurements. Results show that the k − ε model is able to catch the main current propagation parameters (e.g., front velocity at the different phases of the evolution of the current, gravity current depth, etc.), but that a LES model provides more realistic insights into the turbulent processes (e.g., formation of interfacial Kelvin–Helmholtz billows, vortex stretching and eventual break up into 3D turbulence). The ambient fluid velocity field strongly influences the dynamics of the gravity currents. In particular, the presence of an oscillatory motion induces a relative increase of mixing at the front (up to 25%) in proximity of the bottom layer, and further upstream, an increase of the mixing process (up to 60%) is observed due to the mass transport generated by waves. The observed mixing phenomena observed are also affected by the ratio between the gravity current velocity v f and the horizontal orbital velocity induced by waves u w , which has a stronger impact in the wave dominated regime ( v f / u w < 1).
... A variety of turbulence modes are available to solve the RANS equations in FLOW-3D. The three classic turbulence models, namely standard k À ε Harlow and Nakayama (1967), RNG k À ε Yakhot and Smith (1992) and k À ω Wilcox (1998), are compared with the experimental data. Comparisons show that there were no obvious differences between the three turbulence models when simulating the experiments carried out by Ozmen-Cagatay and Kocaman (2010). ...
... It solves the compressible Navier Stokes equations on a 3-D Cartesian FLACS uses RANS (Reynolds therefore a turbulence model is required. For turbulence FLACS uses the k Nakayama [11]. The SIMPLE pressure correction algorithm is applied source terms for the compression work in the enthalpy equation for correct modeling of compressible flows. ...
Releases of hydrogen at elevated pressures form turbulent jets in unconfined & uncongested regions may trigger vapor cloud explosion (VCE) as well as jet fire hazards. In the case of a delayed ignition of an unconfined & uncongested jet the turbulence induced by the jet release can lead to flame speeds sufficient to produce damaging blast loads, even in the absence of confinement or congestion. The VCE hazard posed by such high‐pressure hydrogen releases is not well‐recognized.
The authors have previously presented test data and computational fluid dynamic (CFD) modeling analyses which characterize high pressure hydrogen releases and the associated VCE hazard in unconfined & uncongested regions. The current paper provides an overview of this VCE hazard. It presents both CFD simulations and a new simplified method. The new method determines the blast strength based on a blast curve method (i.e., TNO multienergy or BST) with the strength index correlated to the mass flow rate of the accidental release. The paper discusses also the implications of such events for building siting analyses and risk assessments. © 2018 American Institute of Chemical Engineers Process Saf Prog, 2018
... Hence, the main turbulence model used in this study was the k-ε (RNG) model. The k-ε model is a sophisticated and widely-used model consisting of two transport equations: one for turbulent kinetic energy k and the other for its dissipation ε (Harlow and Nakayama, 1967). The 3D governing equations can be expressed as follows (Launder and Spaulding, 1972): ...
Drought is a natural phenomenon which starts with decreased precipitation and can disrupt the environmental systems by changing the hydrological cycle. This is more conspicuous in hydrological drought. In analysis of hydrological drought, two factors of severity (intensity) and duration play eminent role. These characteristics are highly related and therefore their combined analysis contributes to better understanding of the drought situation. In this research, by using 40-year (1974–2014) daily discharge data of Tajan River, located in Mazandaran province, Iran, and low-flow indices, the best evaluation index of hydrological drought was determined and 10 past hydrological drought events in the region were identified. Then, the best statistical distribution of both drought variables (duration and severity) was selected, based on the goodness-of-fit tests. Five copula functions were fitted to the data. Results showed that Galambos function with the highest maximum log-likelihood (− 8.934) was selected as the best copula function. Results of the bivariate (duration and severity) statistical distribution could be used to analyze the probability of hydrological drought in the region. This bivariate and conditional probability for the worst drought, with duration of 5 months and severity of 0.32, was 6.1 and 28.5%, respectively.
... Hence, the main turbulence model used in this study was the k-ε (RNG) model. The k-ε model is a sophisticated and widely-used model consisting of two transport equations: one for turbulent kinetic energy k and the other for its dissipation ε (Harlow and Nakayama, 1967). The 3D governing equations can be expressed as follows (Launder and Spaulding, 1972): ...
A comprehensive understanding of the sediment behavior at the entrance of diversion channels requires complete knowledge of three-dimensional (3D) flow behavior around such structures. Dikes and submerged vanes are typical structures used to control sediment entrainment in the diversion channel. In this study, a 3D computational fluid dynamic (CFD) code was calibrated with experimental data and used to evaluate flow patterns, the diversion ratio of discharge, the strength of secondary flow, and dimensions of the vortex inside the channel in various dike and submerged vane installation scenarios. Results show that the diversion ratio of discharge in the diversion channel is dependent on the width of the flow separation plate in the main channel. A dike perpendicular to the flow with a narrowing ratio of 0.20 doubles the ratio of diverted discharge in addition to reducing suspended sediment input to the basin, compared with a no-dike situation, by creating the outer arch conditions. A further increase in the narrowing ratio decreases the diverted discharge. In addition, increasing the longitudinal distance between consecutive vanes (Ls) increases the velocity gradient between the vanes and leads to a more severe erosion of the bed, near the vanes.
... The equations for the two turbulence variables, turbulent kinetic energy (k) and vorticity (), are derived using the momentum and continuity equations and time-averaging [41][42]. These equations are presented in the fourth section. ...
In this work, a study involving the third-and fourth-order ENO procedures using the Newton interpolation process from Harten et al. has been presented. The Favre averaged Navier-Stokes equations, in conservative and finite volume contexts, employing structured spatial discretization, are studied. Eleven species chemical models, based on the works of Dunn and Kang, and of Park are considered for the numerical experiments. Turbulence is taken into account considering the implementation of three k-w two-equation turbulence models, based on the works of Coakley; Wilcox; and Yoder, Georgiadids and Orkwis. The " hot gas " hypersonic flow along a blunt body is the numerical experiment for comparisons. The results have indicated that the Coakley turbulence model yields the best prediction of the stagnation pressure value, although the Yoder, Georgiadids and Orkwis turbulence model has presented the best computational performance.
... FLACS is dedicated to the simulation of gas explosions in offshore oil and gas production platforms with high and medium obstruction. FLACS solves the compressible Navier-Stokes equations on a 3-D Cartesian grid using a finite volume method and RANS (Reynolds-Averaged Navier-Stokes) k-ε model for turbulence [11]. The SIMPLE pressure correction algorithm is used [12]. ...
Delayed explosions of accidental high pressure hydrogen releases are an important risk scenario in safety
studies of production plants, transportation pipelines and fuel cell vehicles charging stations. Such
explosions were widely explored in multiple experimental and numerical investigations. Explosion of high
pressure releases in highly obstructed geometries with high blockage ratio is a much more complicated
phenomenon. This paper is dedicated to the experimental investigation of the influence of obstacles on a
delayed deflagration of hydrogen jets. The computational fluid dynamics (CFD) code FLACS is used to
reproduce experimental data. In the current study the computed overpressure signals are compared to the
experimentally measured ones at different monitoring points. Simulations are in close agreement with
experimental results and can be used to predict overpressure where experimental pressure detectors were
saturated. For homogenous stationary clouds a new approach of equivalent mixture of H2/air (~16.5%) to
stoichiometric mixture of CH4/air is suggested. This approach is validated versus experimental data from
the literature in terms of overpressure maxima. A parametric study is performed using FLACS for various
concentrations in the same geometry in order to identify a possible transition from deflagration to
detonation.
... FLACS is dedicated to the simulation of gas explosions in offshore oil and gas production platforms with high and medium obstruction. FLACS solves the compressible Navier-Stokes equations on a 3-D Cartesian grid using a finite volume method and RANS (Reynolds-Averaged Navier-Stokes) k-ε model for turbulence [14]. ...
Explosion venting is a prevention/mitigation solution widely used in the process industry to protect
indoor equipment or buildings from excessive internal pressure caused by an accidental explosion.
Vented explosions are widely investigated in the literature for various geometries, hydrogen/air
concentrations, ignition positions, initial turbulence, etc. In real situations, the vents are normally
covered by a vent panel. In the case of an indoor leakage, the hydrogen/air cloud will be stratified
rather than homogeneous. Nowadays there is a lack in understanding about the vented explosion of
stratified clouds and about the influence of vent cover inertia on the internal overpressure. This paper
aims at shedding light on these aspects by means of experimental investigation of vented hydrogen/air
deflagration using an experimental facility of 1m3 and via numerical simulations using the
computational fluid dynamics (CFD) code FLACS.
... FLACS uses RANS (Reynolds-Averaged Navier-Stokes) approach for fluid mechanics and therefore a turbulence model is required. For turbulence FLACS uses the k-eps model Harlow & Nakayama [11]. The SIMPLE pressure correction algorithm is applied [12]. ...
Releases of hydrogen at elevated pressures form turbulent jets which may pose vapor cloud explosion (VCE) as well as jet fire hazards. The turbulence induced by the jet release can lead to flame speeds sufficient to produce damaging blast loads if the release is not immediately ignited, even in the absence of confinement or congestion. The VCE hazard posed by such high pressure hydrogen releases is not well-recognized.
The authors have previously presented test data and computational fluid dynamic (CFD) modeling analyses which characterize high pressure hydrogen releases and the associated VCE hazard. The current paper provides an overview of this VCE hazard, presents a simplified method for predicting the potential resultant blast loads, and discusses the implications of such events for building siting analyses and risk assessments.
... The Fluid Dynamics community started devoting more time to turbulence modeling after the development of algebraic model of Cebecci and Smith [4] followed by the subsequently the zero, one-equation, and finally the two-equation models of Launder et al. [5], Sarkar and Speziale [6]. Harlow and Nakayama [7,8] in the late 1960s derived the transport equations of eddy viscosity and decay rate of turbulence kinetic energy. Later, Daly and Harlow [9] derived turbulence transport equations describing the flow of incompressible fluid in arbitrary geometry. ...
In this paper, we consider the evolution of decaying homogeneous anisotropic turbulence
without mean velocity gradients, where only the slow pressure rate of strain is non zero. A
higher degree nonlinear return- to - isotropy model has been developed for the slow pressure strain
correlation, considering anisotropies in Reynolds stress, dissipation rate and length scale
tensor. Assumption of single length scale across the flow is not sufficient, from which stems the
introduction of length scale anisotropy tensor, which has been assumed to be a linear function of
Reynolds stress and dissipation tensor. The present model with anisotropy in length scale show
better agreement with well accepted experimental results and an improvement over the Sarkar
and Speziale (SS) quadratic model.
... FLACS uses a k- model in order to model the convection, diffusion, production, and dissipation of turbulence (see, e.g. Harlow & Nakayama, 1967; Launder & Spalding, 1974). However, the standard k- model has been modified by adding source terms for turbulence production by velocity gradients to achieve independent and rapid build-up of the turbulent flow field and representative turbulence production from objects not resolved by the computational grid (subgrid objects). ...
The molten salt fluid-particles transition state is modelled using KTGF-based TFM and CFD-DEM in fluidized beds. The solid phase constitutive equations are modelled using a low density ratio KTGF model. The normal and tangential forces of discrete particles are simulated using CFD-DEM. Two-equation model is used to characterize molten salt fluid turbulence. Wavy-like flow near the bottom and aggregates/fluid voids flow at the upper part coexist along bed height. The molten salt fluid pressure drops and dominant frequencies of wavy-like flow using TFM agree with simulated results using CFD-DEM. The simulated transition state from TFM and CFD-DEM coincides with predictions using four different criteria in literature. The effect of molten salt temperature on normal and tangential Reynolds stresses is discussed. The predicted expansion height and water volume fractions using TFM and CFD-DEM agree with experimental data in an ambient water fluidized bed.
Submarine debris flow, as one of the natural phenomena of marine disasters, often occurs in different sea areas, which has a strong impact on marine structure due to its large specific gravity and high velocity. Submarine pipeline is an important submarine structure which is extremely vulnerable to the marine environment, especially under the submarine debris flow. Many studies have been conducted on the interaction between the debris flow and the pipeline. Based on previous studies, the effects of the debris flows under the water-air interface on the pipeline with degree of freedom are investigated in this study where the debris flow volume, density, slide slope angle, the spring stiffness coefficient, pipeline location and tandem pipeline are considered. According to the computational fluid dynamics theory, a numerical tank is established using the Immersed Boundary (IB) method, in which the three-phase media (air, water and debris flow) are embedded. The results demonstrate that when the debris flow occurs, the water-air free surface is affected by the debris flow propagation, generating a surface wave. When the debris flow passes through the pipeline, with the increase of debris flow density and volume, the force on the pipeline increases, but the vibration frequency of pipeline decreases. Meanwhile, the vibrating pipeline will cause strong turbulence to the debris flow. In addition, the force and the vibration frequency of the pipeline are closely related to the flow pattern of debris flow reaching the pipeline. When the debris flow passes through the tandem pipeline, the closer the distance between the two pipelines is, the more significant the mutual influence between two pipelines is, because the negative pressure area formed behind the upstream pipeline can affect the downstream pipeline to a certain extent.
The hydrodynamic interaction between oblique waves and an arc-shaped breakwater and the wave field behind it. A three-dimensional computational fluid dynamic model was used to simulate the interaction between the oblique waves and arc-shaped breakwater. The pressure distribution and wave force in the different sections under different wave directions were measured by experiments to validate the numerical results. The pressure distribution and wave force in the arc-shaped vertical part of the breakwater along the central axis were further analysed using numerical model. The maximum positive and negative forces in each section along the central axis were compared. The results indicated that the arc curvature exerted little effect on the maximum wave force in the different sections. The wave height behind the breakwater was obviously smaller than that at the front. With the decrease in the incident angle, the influence of diffraction on the wave field gradually decreased. Under east–southeast waves, the maximum wave height behind the breakwater caused by overtopping was approximately 0.7 times the incident-wave height. In the spatial distribution of the wave period behind the breakwater, some areas with smaller periods existed, which may be caused by the overtopping flow that broke behind the breakwater.
The turbulent round jet is considered including the effects of introducing swirl, tabs and rings so as to modify its behaviour. The paper traces the influence of Ricou and Spalding [J Fluid Mech 11(1):21–32, 1961] through almost 60 years of research into this proto-typical engineering and canonically fundamental flow, albeit from a personal perspective. It culminates on current efforts to unravel the turbulent, non-turbulent interface and a brief examination of Spalding’s population modelling ideas, which could assist exploration of this fascinating multi-scale flow.
The turbulent kinetic energy equation is derived and explained in full detail. The motivation and development of RANS-based models is provided, with the aim of generating a deeper understanding of turbulence phenomena. This includes the detailed descriptions for key k-ε, k-ω, and SST hybrid models. Fundamental RANS terms are explained, such as turbulent kinematic viscosity, production, and decay. RANS models are evaluated and compared, and the best overall turbulence model is suggested. Model applicability, best performance regions, and deficiencies are discussed for zero-, one-, and two-equation RANS models. Compelling reasons for avoiding the standard k-ε are provided. Multiple insights regarding ties associated with the development of k-ε and k-ω models are presented, such as the Taylor scale and eddy dissipation.
In this work, a spectral method is applied to the Favre-averaged Navier-Stokes equations in two-dimensions, employing a structured spatial discretization, and using a conservative and finite volume approaches. Turbulence is taken into account considering the implementation of five k-w two-equation turbulence models, based on the works of Coakley 1983; Wilcox; Yoder, Georgiadids and Orkwis; Coakley 1997; and Rumsey, Gatski, Ying, and Bertelrud. The numerical experiments are performed using the Van Leer numerical algorithm. The Euler backward method is applied to march the scheme in time. The spectral method presented in this work employs collocation points and variants of Chebyshev and Legendre interpolation functions are analyzed. Thermochemical non-equilibrium is studied using a five species chemical model. The "hot gas" hypersonic flows around a blunt body, and around a reentry capsule, in two-dimensions, are simulated. The results have indicated that the Chebyshev collocation point variants are more accurate in terms of stagnation pressure estimations. In the blunt body case such errors are inferior to 13.50%, while in the reentry capsule case such errors are inferior to 11.50%, with the best estimations in both problems being inferior to 3.00%. The Legendre collocation point variants are more accurate in terms of the lift coefficient estimations. Moreover, the Legendre collocation point variants are more computationally efficient, although the Chebyshev collocation point variants are cheaper. 1. Introduction There are several approaches for computationally modeling fluid dynamics. These include finite difference, finite element, and spectral methods to name a few. Finite element and finite difference methods are frequently used and offer a wide range of well-known numerical schemes. These schemes can vary in terms of computational accuracy but are typically of lower order of accuracy. If a more accurate solution is desired, it is common practice to refine the mesh either globally or in a region of interest. This can often be a complicated or time consuming process as global mesh refinement will greatly increase the computation time while local refinement requires an elaborated refinement operation [1]. Alternatively, polynomial refinement has been used to improve the solution accuracy and has been shown to converge more quickly than mesh refinement in some cases [2-3]. For finite difference methods, polynomial refinement is performed by including neighboring node values in a higher order polynomial [4]. This can increase the complexity of the scheme especially near the boundaries where nodes do not exist to construct the higher order polynomials. Finite element methods instead increase the number of unknown values within the cell itself to construct a higher order solution [5].
Offshore oil spills can greatly harm marine ecological environments. The containment boom is a simple and effective tool that can effectively prevent the spread of oil spills, reduce the oil-spreading region, and work with other recovery measures. In the study described, a two-dimensional multiphase model was developed to reproduce the oil containment process and to comprehensively investigate the water–oil boom interaction. Good agreement was achieved between the modeled and measured results in an application of the oil containment process with respect to variation in oil thickness. The verified model was then employed to investigate oil containment subject to waves and currents. In addition, the containment performance of floating booms with different skirt lengths, buoyancy/weight ratios, and boom shapes was investigated. The numerical results showed that boom motion can significantly change the flow field around the boom and thereby the boom’s containment effectiveness. Booms with arc-shaped or polyline-shaped skirts consistently exhibit satisfactory oil-controlling performance.
This paper presents systematic comparisons of dam-break wave generation and propagation between numerical solutions of RANS equations with three different turbulence models (standard k – ε, RNG k – ε and k – ω), numerical solutions of Euler equations, analytical solutions and experimental data from literature for both dry-bed and wet-bed cases. Results show that numerical solutions of RANS equations with the three different turbulence models can accurately calculate the whole process of dam-break wave propagating over both dry-bed and wet-bed. Numerical solutions of Euler equations show remarkable discrepancies compared with the experimental data in dry-bed condition, but can predict the dam-break wave well in wet-bed conditions. The merits and demerits of the calculation methods are discussed in full and their application scopes are suggested.
Purpose
Biological pharmaceutical unit operations like homogenization or pooling of liquids are often performed in stirred vessels. Bottom-mounted magnetic stirrers are usually the system of choice in drug product manufacturing, because bottom-mounted magnetic stirrers are considered to be gentle mixing systems. Nevertheless, magnetic stirrers can cause shear stress and, thus, lead to protein damage.
Methods
This study uses computational fluid dynamics (CFD), because flow and shear rates cannot easily be measured at the spot of interest. The investigation utilizes CFD models, which were checked for plausibility by comparing experimental results and model outcome. The investigators first modeled macroscopic flow across a range of vessel volume capacities. Subsequently, detailed models focusing on two locations (bearing gap (2 mm - 3.5 mm) and spigot gap (40 μm - 80 μm)) were developed.
Results
The macroscopic flow modeling showed that the direction of flow varies based on the vessel volume capacity. The detailed CFD model estimated significant flow through the bearing gap. However, the calculated shear rates in the bearing gap were always lower than the shear rates which occur directly next to the impeller tip. The CFD model calculated significantly higher shear rates in the spigot gap and flow in the lower microliter range.
Conclusions
Shear rates at the impeller tip are typically used as parameter to characterize stirred mixing systems. Although higher shear rates were found in the spigot gap, these higher shear rates can most likely be neglected for most applications due to non-significant flow through the spigot gap.
Accurate and robust models for the pressure strain correlation are an essential component for the success of Reynolds Stress Models in turbulent flow simulations. However replicating the non-local action of pressure using only local tensors places a large limitation on potential model performance. In this thesis we outline an approach that extends the tensor basis used for pressure strain correlation modeling to formulate models with improved precision and robustness. This set of additional tensors is analyzed and justified based on physics based arguments and analysis of simulation data. Using these tensors models for the rapid and slow pressure strain correlation are developed. The resulting complete pressure strain correlation model is tested for a wide variety of turbulent flows, while being contrasted against the predictions of established models. It is shown that the new model provides significant improvement in prediction accuracy. In the second stage of the work a series of experiments on decaying grid generated turbulence and grid turbulence with mean strain were conducted. Experimental data of turbulence statistics including Reynolds stress anisotropies is collected, analyzed and then compared to the predictions of Reynolds Stress Models to assess their accuracy and is used to evaluate the variability in the coefficients of the rate of dissipation model and the pressure strain correlation models used in Reynolds Stress Modeling.
In this work, a study involving the Maciel scheme to solve the reactive Favre averaged Navier-Stokes equations, coupled with a turbulence model and the Maxwell equations is performed. The Favre averaged Navier-Stokes equations coupled with the Maxwell equations, in conservative and finite volume contexts, employing structured spatial discretization, are studied. Eleven species chemical models, based on the works of Dunn and Kang and of Park are considered for the numerical experiments. Turbulence is taken into account considering the implementation of five k-w two-equation turbulence models, based on the works of Coakley 1983; Wilcox; Yoder, Georgiadids and Orkwis; Coakley 1997; and Rumsey, Gatski, Ying and Bertelrud. For the magnetic coupling, the Gaitonde formulation is taken into account. The "hot gas" hypersonic flow around a blunt body is the numerical experiment for comparisons. The results have indicated that the Maciel scheme using the Dunn and Kang chemical model coupled with the Coakley 1983 turbulence model yields the best prediction of the stagnation pressure value and the best prediction of the lift coefficient. On the other hand, the Maciel scheme using the Park chemical model coupled with the Coakley 1997 turbulence model is more computationally efficient. Errors in the stagnation pressure estimation inferior to 5.00% were found.
The technique of Computational Fluid Dynamics (CFD) was used to study the indoor airJowfield in mechanically ventilated and air-conditioned spaces. Numerical experiments were performed in six test sites of different sizes and ventilation configurations. Typical mechanical ventilation or airconditioning systems were installed in those sites so that the evaluation of results of the system performance would be usefulfor assessing other similar systems. The predicted results were analysed and compared with experimental measurements. More importantly, the potential use of the predicted results was illustrated. Correlation relationships between the flow field characteristics and the macroscopic design parameters were established.
A numerical method has been used to solve completely the full equations for the motion of a viscous, heat-conducting, incompressible fluid flowing through a channel past a rectangular cylinder. The dynamics of the unsteady flow were made visible by means of a number of different types of contour plots, and selected ones are presented and discussed to illustrate various newly discovered features of the von Ka´rma´n vortex street. In addition, the heat transfer was measured from the heated rear of the cylinder to the colder fluid, with data presented in the form of Nusselt number correlated with Reynolds number. Corrections are derived for blockage, and Prandtl number scaling is discussed.
A kinetic model of the final‐stage decay of grid‐produced turbulence is presented. Theoretical analysis leads to an inverse‐square energy decay law. The prediction has been confirmed by measurements in a low‐speed water channel. A replot of the Dryden‐Schubauer and Batchelor‐Townsend wind‐tunnel data also indicates better agreement with the present inverse‐square, rather than the‐power decay law proposed by earlier investigators.
A discussion is given concerning the effects of viscosity and large disturbance amplitude on the instability occurring at a slip interface between two parts of an incompressible fluid. In particular it is shown how an analysis by Carrier can be applied to explain the train of small vortices generated at a shearing edge, and comparisons with experimental results are given for a number of examples. In addition, the numerical method of Fromm has been applied to obtain complete solutions of the slip instability problem, and numerous results are presented to show additional comparisons with the linearized theory for viscous fluids, and to show in detail the structure of the late-stage nonlinear flow in which the vortex sheet may tend to ``roll up.''
A method is described for the solution of time-dependent problems concerning the flow of viscous incompressible fluids in several space dimensions. The method is numerical, using a high-speed computer for the solution of a finite-difference approximation to the partial differential equations of motion. The application described here is to a study of the development of a vortex street behind a plate which has impulsively accelerated to constant speed in a channel of finite width; the Reynolds-number range investigated was 15 ≤ R ≤ 6000. Particular attention was given to those features for which comparison could be made with experiments, namely, critical Reynolds number for vortex shedding, drag coefficient, Strouhal number, vortex configuration, and channel-wall effects. The nature of the early stages of flow-pattern development was also investigated.
A mechanism is proposed for the manner in which the turbulent components support Reynolds stress in turbulent shear flow. This involves a generalization of Miles's mechanism in which each of the turbulent components interacts with the mean flow to produce an increment of Reynolds stress at the ‘matched layer’ of that particular component. The summation over all the turbulent components leads to an expression for the gradient of the Reynolds stress τ(z) in the turbulence
\[
\frac{d\tau}{dz} = {\cal A}\Theta\overline{w^2}\frac{d^2U}{dz^2},
\]where is a number, Θ the convected integral time scale of the w-velocity fluctuations and U(z) the mean velocity profile. This is consistent with a number of experimental results, and measurements on the mixing layer of a jet indicate that A = 0·24 in this case. In other flows, it would be expected to be of the same order, though its precise value may vary somewhat from one to another.
Auf Grund experimenteller Untersuchungen des turbulenten Austausches wurde eine Formel für die Geschwindigkeit turbulenter Strömungen in Rohren und Kanälen entwickelt. Diese Formel stellt die Geschwindigkeit über einem Halbmesser vollständig dar, d. h. sie gilt näherungsweise sowohl für die unmittelbare Wandnähe als auch für den mittleren Teil des Strömungsquerschnittes.
From experimental investigations of turbulent exchange of momentum a formula is derived for the velocity of turbulent flow in tubes and pipes. This formula gives a complete representation of the velocity distribution, i. e. it holds good close to the wall as well as in the core of the flow.
A new technique is described for the numerical investigation of the time‐dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free. The full Navier‐Stokes equations are written in finite‐difference form, and the solution is accomplished by finite‐time‐step advancement. The primary dependent variables are the pressure and the velocity components. Also used is a set of marker particles which move with the fluid. The technique is called the marker and cell method. Some examples of the application of this method are presented. All non‐linear effects are completely included, and the transient aspects can be computed for as much elapsed time as desired.