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Chapter 1: Introduction
Sumps

Sump is a depressed place where liquid collects and prepare to start processing. In several plants, cooling water is set in sump to be withdrawn inside the plant using electric pumps. Sumps usually have a lower level than the rest of facility that it serves. That means sumps in most cases need a powerful pump or a complete pumping station in order to withdraw large amount of liquid (typically water) in short time to achieve the required purpose. Thus large amount of water is moving in and out of sump in short time and in some cases this process can be repeated for a long time and for multiple times.
Sumps have many shapes. However, the most common shapes are the circular and the rectangle shape. Inside of some sumps, there may be also some sort of structures like spillways, wires, and walls, which play a role of flow controlling structures inside sump.
The main role of sump is to collect liquid form its source. These sources can be pumped liquid, lakes or rivers or any other liquid sources. According to the working conditions, the sump should be designed in order not to run out of water or get flooded during operation.

Figure 1.1: Domestic sump pump example

Vortices

In fluid dynamics, a vortex is a region in a fluid domain in which the flow rotates around an axis line, which may be straight or curved. The particles movement’s pattern may be used to classify the vortex. In 2D perspective, the fluid particles can move in rotational or irrational motion flow. However, particles’ movement is not the most common point of interest in industrial application of science, So new classifications are developed to help studying vortices in depth.
There are many types of vortices that are classified based on their location and intensity. The main two categories of vortices are “Free Surface” and “Sub-Surface” types. As shown in figure(1.2) there are six types of free surfaces vortices from type 1 to type 6 according to their intensity. When intensity of vortex increases, it tries to entrap air inside the water. In type 6 vortices, a full cone of air is developed inside intake and air find its way to the suction pipe. Vortices also pull floating trash form the surface to the intake that makes the liquid not clear and increase the chance for blocking the suction pipe or damage the pump.

Figure 1.2: Free surface and sub-surface vortices classification

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Problem definition

In particle life, Sumps play main role in supplying facilities with water via suction pumps. This is where the vortices appear. From economic perspective, the sump should be as small as possible to save both money and place. To achieve that point of view, the sump must pass a large amount of fluid at high velocity and that lead to massive disturbance in flow inside sump. Disturbed flow does not move in smooth way as predicted inside the sump. Instead, the flow moves in random directions at high speed. As a result, flow starts to make vortices inside the domain of flow. Vortices can cause major issues to the suction pump. It can pull trash and air form water surface to the water inside the sump. Pulled air and trash reduce the quality of water inside the sump. In addition, the trash can block the suction pipe or damage the pump impeller. Trapped air can also cause cavitation inside suction pipe or inside the pump, leading to critical damages on them and increase the rate of maintenance necessity.
Although water disturbance around the inlet or outlet is considered disadvantage of pump sump, there is also another case where water stood stagnant inside the sump. This happens when bad arrangement of inlet/outlet pipes is used. As result, large amount of water may be forced to hold still and not participate in the pumping operation flow. Which is not a desired behavior in water sumps, especially for fresh water sumps. The stagnant water gets its quality reduced by time and highly increases the need for sump cleaning.
In all cases, pumping efficiency is deeply reduced as a result of the flow pattern in the sump. Which is the main disadvantage of vortices and disturbed flow. In order to eliminate this issues, engineers have find out that sump should be hydraulically designed to achieve a non-disturbed flow inside sump. However, there are no equation is developed to solve this problem. Instead, engineers used to build a scaled model of sump in laboratory and test it against several predefined hypothetical predicted scenarios and configurations. Building scaled model is both time and money consuming, which both are not significantly available in projects design phase. In addition, it may not be helpful to try multiple solutions and configurations to solve one disturbance problem in the sump.
After a while, design recommendations had been developed based on many previous laboratory and field experience. The design of sump uses a set of parameters that controls the flow in sump. However, these design recommendations cannot solve all types, shapes and configurations of sumps and those outlier sumps must be lab modeled to be studied accurately.
Nowadays computers had break into the science field of fluid dynamics, and a new approach of solving fluid motion have appeared in applied fluid dynamics filed. The solution accuracy is based on numerical model applied and boundary conditions used to represent the domain. As a result of using computer in modeling flow, now one can run simulations and obtain results just like the laboratory tests and it can be done as many times as it takes to accurately study a sump case. However, the more accurate the result is required the more time and processing power is required.

Objective of study

The main objective of this research is to study the flow disturbance and vortices in size pump sumps that are considered relatively small to medium in size and have a discharge less than 1.0 m3/s. This objective shall be reached by simulating the flow inside pump sumps using suitable CFD code (Ansys Fluent) modified for such purpose. These simulations are depending on parameters affect the ordinary design recommendations based on previous practice and experimental knowledge. They are also measure the flow disturbance and the effect of each parameter on the flow. The study also tries to find the sensitivity of each parameter on flow disturbance in pump sump and explorer the modifications that tries to solve or reduce sump problems.

Thesis outlines

The thesis consists of five chapters discussing the following:
Chapter One: presents introductory objects required to be known before starting exploring the thesis such as identification of sumps and vortices. It also contains problem definition and thesis objective.
Chapter Two: includes literature reviews of previous experimental and numerical studies for flow in pump sumps and previous purposed solutions for disturbed flow in pump sump.
Chapter Three: describes the mathematical model, numerical approach solution and boundary conditions used for simulating flow in CFD software (Fluent).
Chapter Four: includes the list of parameters used in simulations and the defined cases to be run. It also includes the methods for assessing and verifying the model’s runs. At the end of the chapter some refined cases are generated based on previous simulations results and the refined simulations’ results are discussed.
Chapter Five: includes the conclusion of the study and discuss the results of the simulations. It also contains the future work and thesis recommendation for sump design.
Appendix A: includes detailed sketches for geometries of sumps used to perform the simulations.

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Chapter 2: Literature Review
2.1 Previous experimental and empirical researches.

Upon discovering the importance of pump’s sump flow patterns, many researchers have been trying to find out the way the flow act in sump. As the flow regime is not quite typical, it is a hard quest to predict the flow pattern in sumps. However, Researchers tried to figure out how the flow behave as follow.

Mahadevan Padmanabhan and George E.Hecker (1984), investigated the scale effect of laboratory pump sump models(geometric scales of 1:2 and 1:4 ) and determine whether scaling may distort the predictive ability of hydraulic models of pump sumps with such large scale ratios. Possible scale effects on vortexing, pipe swirl, inlet losses, and air ingestion due to air?drawing vortices have been examined using the experimental results. This research led to discovery that there are no significant scale effects on free?surface vortexing were evident in the tested ranges.

A. Jacob Odgaard and James J. Dlubac (1984), at the Iowa Institute of Hydraulic Research, reported the results of hydraulic model study of proposed design of a 4-pump sump for withdrawal of cooling water. The results of the study suggest that potential-flow theory can be used as a guide toward the development of sump design modifications and even may be used in the development of the conceptual layout of pump sumps in the future.

James A. Liggett (1990), have noticed that computer have broken into hydraulics engineering and become an everyday tool in hydraulic practice. He advised to reduce the gap between research and practice and emphasized on the importance for hydraulic engineer to know when to solve a problem using computer and when it is wrong to do this.

Gustavo Arboleda , Mutasem El-Fadel (1996), investigated one obstacle to the development of a generalized sump design, which is the geometry and alignment of the areas immediately upstream of the sumps. They studied one particular sump design, developed through hydraulic model test, which was highly dependent on approach flow conditions. The test concluded that major changes are required to be done, in order to generalize the current design practice.

American national standards institute (1998), released the “American National Standards for Pump Intake Design ANSI/HI 9.8-1998”. This standard was developed from the collected experiences from practical fieldwork and experimental data form public researches. The aim of standards is to eliminate the misunderstanding between the manufactures, purchasers and users. It also helps hydraulics’ engineers in design phase and provide a guide line for a more stable design based on previous experience available.

Matahel Ansar and Tatsuaki Nakato (2001), carried out an experimental study to obtain detailed measurements of three-dimensional turbulent flows within a rectangular single pump bay area using an Acoustic Doppler Velocimeter (ADV) in order to elucidate swirling flow characteristics within the pump sump. Uncertainty analysis for ADV velocity measurements showed good quality of collected data, with uncertainty in mean velocities varying from 2.5 to 6.4%. These experimental data were used in validating inviscid numerical solutions.
Hamed Sarkardeh , Amir Reza Zarrati & Reza Roshan (2010), studied the effect of the intake head wall slope and the installation of a trash rack on the type and strength of vortices. The study was carried out experimentally. The strength within each vortex was determined by measuring its tangential velocity utilizing an Acoustic Doppler Velocimeter (ADV). Experiments were carried out with a projected intake and an intake with various head wall slopes, discharges and submerged depths. The results from the projected intake tests indicated that “type 6” vortex was present. However, the vortex strength and type reduced as the intake head wall slope increased up to the vertical position. In addition, increasing the intake head wall slope increased the vortex instability.

Kerem Ta?tan & Nevzat Yildirim (2013), investigated the importance of using both the kinematic and dynamic similarities in scaled models of the critical submergence of vertical vortices at intake structures. They used semi-theoretical approach, which is based on the principle of flow continuity in combination with published experimental data. The new finding was that dynamic similarity is not required for the prediction of the ratio of the critical submergence to the intake diameter.

Frank Suerich-Gulick, Susan. J. Gaskin, Marc Villeneuve & Étienne Parkinson (2014), ran a laboratory experiment for hydropower facility scaled model. These models were used for assessment of vortex formation during the design phase. Proposes a semi-empirical model that roughly predicts how the approach flow and intake geometry determine the key vortex characteristics (the core radius, bulk circulation and the depth of the free surface depression). The model was developed using detailed velocity measurements of the approach flow and the flow inside the vortex in a laboratory-scale physical model, using analytical models and insights drawn from previous work.
They expended their study by investigating the impact of surface tension, viscosity and turbulence on the scaling behavior of the vortices examined here using an analytical free surface vortex model developed from measurements in a laboratory-scale hydropower intake. They examined the effect of surface tension on the free surface depression using a finite-difference model over a wide range of depression scales and shapes. The impact and scaling behavior of surface tension are found to be qualitatively different depending on whether the depression is dimple- or funnel-shaped. The effect of viscosity on scaling predicted by the analytical vortex model disagreed trends recorded by previous authors, which lead to the fact that additional processes such as turbulent diffusion may play a significant role at larger scales.
They also ran an experiment to investigate the effect of intake-entrance profiles on free-surface vortices in pump sump. The focus of the study was on a range of Froude numbers from 0.25 to 0.65. The square-edged shape appeared to show the highest local head-loss compared to other shapes. Steady counter-clockwise vortices characterize all the intake profiles except in a narrow water tank. The results demonstrate that the intake-entrance profile has an important effect on the critical submergence.

Anton de Fockert, Jos M.C. van ‘t Westende ; Femke I.H. Verhaart (2015), found out that the pump intake design standard ANSI/HI 9.8-2012 didn’t define the measurement techniques of swirls or pre- swirls, thus they ran lab experiments using two scale models where swirl was measured for 6 hours. They also noticed that the ANSI/HI 9.8-2012 standard didn’t define strict acceptance criteria for future sump model tests proposed.

Hongliang Sun ; Yakun Liu (2015), ran an experiment with a free surface vortex in a cylindrical tank. The vortex flow field was measured using particle image velocimetry (PIV) and extensive experimental data describing the vortex characteristics (radial and tangential velocity distributions, vortex core radius variation, water surface profile and circulation distribution) have been obtained and analyzed. Based on the experiment data, they developed an empirical model describing the key vortex characteristics. In addition, using the pressure distributions inside and outside the vortex core, a dimensionless equation for the critical submergence requiring only the Froude number and the circulation number is derived. Results using this equation are in agreement with the experimental data.

2.2 Numerical researches.

In late nineties world of hydraulic engineers started to pay attention of computer usage helping solve difficult hydraulic problems. Using scientific approach, they started to try to simulate the problems and then assessment and validate the results then get conclusions about problem.

John E. Hite and Walter C. Mih (1994), developed concise equations by modifying the equation for tangential velocity originally proposed by Rosenhead in 1930. Laboratory experiments were conducted on strong air?core vortices near a water intake. The results indicate the equations agree with experimental measurements and are applicable to vortex motion in general

G. S. Constantinescu,V. C. Patel (1998), developed a computational fluid dynamics model to simulate the three-dimensional flow field in a pump intake and study the formation of vortices in sump. the model solved the he Reynolds-averaged Navier-Stokes equations and two turbulence-model equations. because the case study was selected according to commonly used design criteria, that lead to weak vortices to be expected. The study suggests that the numerical model should be used with other geometrical arrangements and flow parameters.

V. S. Neary, F. Sotiropoulos and A. J. Odgaard (1999), developed a three-dimensional (3D) numerical model for predicting steady, mean turbulent flows through lateral intakes with rough walls. The method solves the Reynolds-averaged Navier-Stokes equations closed with the isotropic k-? turbulence model.

Liang Ge and Fotis Sotiropoulos (2005), introduced a general-purpose numerical method for solving the full three-dimensional (3D), incompressible, unsteady Reynolds-averaged Navier-Stokes (URANS) equations in natural river reaches containing complex hydraulic structures at full-scale Reynolds numbers. The URANS and turbulence closure equations are discretized using a second-order accurate finite-volume approach. The discrete equations are integrated in time via a dual-time-stepping, artificial compressibility method in conjunction with an efficient coupled, block-implicit, approximate factorization iterative solver. The computer code is parallelized to take full advantage of multiprocessor computer systems so that unsteady solutions on grids with 106 nodes can be obtained within reasonable computational time. The power of the method is demonstrated by applying it to simulate turbulent flow at R?107 in a stretch of the Chattahoochee River containing a portion of the actual bridge foundation located near Cornelia, Georgia. It is shown that the method can capture the onset of coherent vortex shedding in the vicinity of the foundation while accounting for the large-scale topographical features of the surrounding river reach.

Liang Ge, Seung Oh Lee, Fotis Sotiropoulos and Terry Sturm (2005), presented a model for solving the unsteady Reynolds-averaged Navier-Stokes (RANS) equations in arbitrarily complex, multi-connected domains. The measured mean velocity and turbulence kinetic energy profiles are compared with the numerical simulation results and are shown to be in good agreement with the numerical simulations. A series of numerical tests is carried out to examine the sensitivity of the solutions to grid refinement and investigate the effect of inflow and far-field boundary conditions. As further validation of the numerical results, the sensitivity of the turbulence kinetic energy profiles on either side of the complex pier bent to a slight asymmetry of the approach flow observed in the experiments is reproduced by the numerical model.

T. E. Tokyay and S. G. Constantinescu (2006), developed a reliable numerical model to predict pump intake flow and associated vortices. In their work, large-eddy simulation (LES) in conjunction with an accurate non-dissipative non-hydrostatic Navier-Stokes massively parallel solver is used to predict the flow and vortical structures in a pressurized pump intake of complex geometry. The LES model is validated using particle image velocimetry data recently collected on a laboratory model of a realistic geometry pump intake. Overall, LES can more accurately predict the mean flow and turbulence statistics compared to the steady Shear Stress Transport (SST) model.

Liaqat A. Khan, Edward A. Wicklein and E. C. Teixeira (2006), presented A three-dimensional (3D) computational fluid dynamics (CFD) model of a contact tank. The objective was to demonstrate that CFD models can simulate both the flow through curve (FTC) data and the 3D velocity field quite well. The physical model’s studies indicate that different baffle arrangements can lead to similar FTCs. Therefore, a good prediction of only FTC, as presented in previous 3D CFD model studies, does not necessarily imply a correct simulation of the flow field.

Akihiko Nakayama & Nobuyuki Hisasue (2010), had simulated numerically three-dimensional unsteady flow in the intake channel of a small-scale hydroelectric power facility. The numerical method used in simulation is the highly simplified MAC iteration method modified for the free-surface flow. the complex and unsteady nature of the vortical flow can be reproduced well. The study shows that this method is relevant for predicting the intake flows with possible air entrainment.

Yingkui Wang , Chunbo Jiang & Dongfang Liang (2011), carried out a comparison between empirical formulae of intake vortices. Based on the axisymmetric Navier–Stokes equations and empirical assumptions, two sets of formulations for the velocity distributions and the free surface profiles are proposed and validated against measurements available in the literature. Compared with previous formulae, the modifications based on Mih’s formula are found to greatly improve the agreement with the experimental data. Physical model tests were also conducted to study the intake vortex of the Xiluodu hydroelectric project in China. A good agreement was again observed between the prediction and the measurements.

Quentin B. Travis and Larry W. Mays (2011), used the nearest neighbor’s algorithm on available experimental and field data to develop a formal database for equivalent of the data proximity procedure. This database was partitioned and the machine learning parameters adjusted to obtain a stochastic model with maximum predictive accuracy. They discovered that the geometry approach doesn’t have a significant effect on model. probability charts generated from the model show regions of vortex formation and problems more numerous and larger on average than regions of low vortex probability.

Zachary H. Taylor, Jeremy S. Carlston ; Subhas Karan Venayagamoorthy (2015), focused their study on understating the hydraulic design of baffled contact tanks using computational fluid dynamics simulations. They dedicated their search to answer the key question: “for a given footprint of a rectangular tank with a specified inlet width (Winlet), how does the hydraulic efficiency of a baffled tank depend on the configuration of internal baffles? “. In the way of solving this question they ran 30 carefully chosen scenarios for 2D planner to quantify the hydraulic efficiency of a laboratory scale tank as a function of dimensional relationships between key baffle design dimensions (baffle opening length Lbo, baffle channel width Wch, and baffle channel length LT). Simulated longitudinal velocity profiles and flow through curves show good agreement with previous experimental results. The results indicate that the hydraulic efficiency can be optimized by ensuring that Lbo/Wch ? 1 and orienting baffles along the longer direction of the rectangular footprint.

Fotis Sotiropoulos (2015), noticed the exponentially growing computing power in the last era and the developing of a powerful simulation-based engineering science framework, which encouraged him to review such progress and offer specific examples highlighting the enormous potential of simulation-based engineering science to supplement and dramatically augment the insights that can be gained from physical experiments. He showed that now we can run a Multi-physics simulations taking into account complex waterway bathymetry, energetic coherent structures, turbulence/sediment interactions and morphodynamics, free-surface effects and flow structure interaction phenomena and the advantage of the growing computer power.

Z. W.Guo, F. Chen, P. F. Wu and Z. D. Qian (2017), investigated the three-dimensional unsteady vortex flow in a sump pump using computational fluid dynamics (CFD) method with the shear-stress transport (SST) k?? model and the volume of fluid (VOF) multiphase model. The helicity density, involving both velocity and vortex, is especially used to discuss the mechanism of the process for air entrainment. The calculated results are qualitatively validated compared with the vortex observed in a previous experiment. The results investigated show that the helicity density takes a very important role in forming the process of the air entrainment. When the dimple appears, the helicity density near the intake pipe is larger than that close to the surface. When the air begins to flow into the pipe, the helicity density tube close to the surface reaches a certain level to connect the helicity density tube from the intake pipe. Moreover, the air entrainment can be suppressed if the helicity density tube is broken by placing a circular plate in a certain position.

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Chapter 3: Numerical Modeling
3.1 Introduction to CFD

Since the beginning of using computers in hydraulics, engineers started to solve some very complicated problems involving multi-physics systems, turbulence, fluid dynamics, etc. Hence, a new term had appeared in hydraulic science which is “CFD” that refers to “Computational Fluid Dynamics”. This branch of hydraulic science is interested in studying flow behavior and interactions between different types of fluids and gasses using numerical methods. It can analysis the flow and simulate fluids including turbulence, sediment and energy. CFD also have some non-hydraulic uses like arterial blood flow simulation, aerodynamic studies in planes and other vehicles.

3.2 Discretization methods

CFD developed codes can find solution for problems numerically using one of the following Discretization methods, and commercial application usually gives the choice of using any of them as required:
Finite Volume Method
The Finite Volume Method (FVM) is a common approach for solving used in CFD codes. It has advantages in large problems especially problems with high Reynolds number and turbulent flows, and source term dominated flows (like combustion).
The governing partial differential equations are the Navier-Stokes equations, the mass and energy conservation equations, and the turbulence equations. When they are solved over discrete control volumes. This discretization guarantees the conservation of fluxes through a particular control volume. The finite volume equation yields governing equations in the following form:
?/?t ???QdV+ ???FdA=0??
Where “Q” is the vector of conserved variables,” F” is the vector of fluxes, “V” is the volume of the control volume element and “A” is the surface area of the control volume element.

Finite element method
The Finite element method(FEM) has wide range of application in structure analysis filed. It is also applicable for both solid and fluid analysis. Although FEM must be carefully formulated to be conservative, it is much more stable than the finite volume approach. But also it requires more resources and computer power to obtain conservative solutions with high resolution. This method uses the following weighted residual equation:
R_i= ???W_i QdV^e ?
Where “Ri” is the equation residual at an element vertex i, “Q” is the conservation equation expressed on an element basis, “Wi” is the weight factor, and “Ve” is the volume of the element.

Finite difference method
This method has historical usage as it is considered an easy method to implement in a program. It is currently used in few applications specialized in handling complex geometry or overlapping mesh where the solution is interpolated between overlapped meshes. The primary equation of this method is:
?Q/?t+?F/?x+?G/?y+?H/?z=0
Where “Q” is the vector of conserved variables, and “F”, “G”, and “H” are the fluxes in the x, y, and z directions respectively.

Spectral element method
Spectral element method is a finite element type method. It requires the mathematical problem (the partial differential equation) to be cast in a weak formulation. This is typically done by multiplying the differential equation by an arbitrary test function and integrating over the whole domain. Purely mathematically, the test functions are completely arbitrary – they belong to an infinite-dimensional function space. Clearly an infinite-dimensional function space cannot be represented on a discrete spectral element mesh; this is where the spectral element discretization begins. The most crucial thing is the choice of interpolating and testing functions. In a standard, low order FEM in 2D, for quadrilateral elements the most typical choice is the bilinear test or interpolating function of the form “v(x,y)=ax+by+cxy+d”. In a spectral element method however, the interpolating and test functions are chosen to be polynomials of a very high order (typically e.g. of the 10th order in CFD applications). This guarantees the rapid convergence of the method. Furthermore, very efficient integration procedures must be used, since the number of integrations to be performed in a numerical code is big. Thus, high order Gauss integration quadratures are employed, since they achieve the highest accuracy with the smallest number of computations to be carried out. At the time there are some academic CFD codes based on the spectral element method and some more are currently under development, since the new time-stepping schemes arise in the scientific world.

Each of the previous method have some advantages/disadvantages according to objective of the code and required solution. For example, finite volume method (FVM) has great advantage in memory usage and solution speed specially for large problems. This is why it is very common for CFD codes designed to be running on small and low speed computers.

Figure 3.1: Simulation of blood flow in a human aorta artery
3.3 Turbulence

In order to have an accurate simulation of fluid, turbulence must be taken into account. This can be done be using turbulence model in combination with the previous motioned models. In computational modeling of turbulent flows, one common objective is to obtain a model that can predict quantities of interest, such as fluid velocity, for use in engineering designs of the system being modeled. For turbulent flows, the range of length scales and complexity of phenomena involved in turbulence make most modeling approaches prohibitively expensive; the resolution required to resolve all scales involved in turbulence is beyond what is computationally possible. The primary approach in such cases is to create numerical models to approximate unresolved phenomena.
Turbulence models can be classified based on computational expense, which corresponds to the range of scales that are modeled versus resolved (the more turbulent scales that are resolved, the finer the resolution of the simulation, and therefore the higher the computational cost). If a majority or all of the turbulent scales are not modeled, the computational cost is very low, but the tradeoff comes in the form of decreased accuracy.

In addition to the wide range of length and time scales and the associated computational cost, the governing equations of fluid dynamics contain a non-linear convection term and a non-linear and non-local pressure gradient term. These nonlinear equations must be solved numerically with the appropriate boundary and initial conditions.

Reynolds-averaged Navier–Stokes

Reynolds-averaged Navier–Stokes (RANS) equations are the oldest approach to turbulence modeling. A set of the governing equations is solved, which introduces new apparent stresses known as Reynolds stresses. This adds a second order tensor of unknowns for which various models can provide different levels of closure. It is a common misconception that the RANS equations do not apply to flows with a time-varying mean flow because these equations are ‘time-averaged’. In fact, statistically unsteady (or non-stationary) flows can equally be treated. This is sometimes referred to as URANS. There is nothing inherent in Reynolds averaging to preclude this, but the turbulence models used to close the equations are valid only as long as the time over which these changes in the mean occur is large compared to the time scales of the turbulent motion containing most of the energy.

RANS models can be divided into two broad approaches:

Boussinesq hypothesis
This method involves using an algebraic equation for the Reynolds stresses which include determining the turbulent viscosity, and depending on the level of sophistication of the model, solving transport equations for determining the turbulent kinetic energy and dissipation. Models include k-? (Launder and Spalding) Mixing Length Model (Prandtl) and Zero Equation Model (Cebeci and Smith). The models available in this approach are often referred to by the number of transport equations associated with the method. For example, the Mixing Length model is a “Zero Equation” model because no transport equations are solved; the k-? is a “Two Equation” model because two transport equations (one for k and one for ?) are solved.

Reynolds stress model (RSM)
This approach attempts to actually solve transport equations for the Reynolds stresses. This means introduction of several transport equations for all the Reynolds stresses and hence this approach is much costlier in CPU effort.

Direct numerical simulation

Direct numerical simulation (DNS) resolves the entire range of turbulent length scales. This marginalizes the effect of models, but is extremely expensive. The computational cost is proportional to R_e^3. DNS is hard to apply for flows with complex geometries or flow configurations.

Large eddy simulation

Large eddy simulation (LES) is a technique in which the smallest scales of the flow are removed through a filtering operation, and their effect modeled using subgrid scale models. This allows the largest and most important scales of the turbulence to be resolved, while greatly reducing the computational cost incurred by the smallest scales. This method requires greater computational resources than RANS methods, but is far cheaper than DNS.

Detached eddy simulation

Detached eddy simulations (DES) is a modification of a RANS model in which the model switches to a subgrid scale formulation in regions fine enough for LES calculations. Regions near solid boundaries and where the turbulent length scale is less than the maximum grid dimension are assigned the RANS mode of solution. As the turbulent length scale exceeds the grid dimension, the regions are solved using the LES mode. Therefore, the grid resolution for DES is not as demanding as pure LES, thereby considerably cutting down the cost of the computation. Though DES was initially formulated for the Spalart-Allmaras model (Spalart et al., 1997), it can be implemented with other RANS models (Strelets, 2001), by appropriately modifying the length scale which is explicitly or implicitly involved in the RANS model. So while Spalart–Allmaras model based DES acts as LES with a wall model, DES based on other models (like two equation models) behave as a hybrid RANS-LES model. Grid generation is more complicated than for a simple RANS or LES case due to the RANS-LES switch. DES is a non-zonal approach and provides a single smooth velocity field across the RANS and the LES regions of the solutions.

Coherent vortex simulation

The coherent vortex simulation approach decomposes the turbulent flow field into a coherent(consistent) part, consisting of organized vortical motion, and the incoherent part, which is the random background flow. This decomposition is done using wavelet filtering. The approach has much in common with LES, since it uses decomposition and resolves only the filtered portion, but different in that it does not use a linear, low-pass filter. Instead, the filtering operation is based on wavelets, and the filter can be adapted as the flow field evolves. Farge and Schneider tested the CVS method with two flow configurations and showed that the coherent portion of the flow exhibited the -40/39 energy spectrum exhibited by the total flow, and corresponded to coherent structures (vortex tubes), while the incoherent parts of the flow composed homogeneous background noise, which exhibited no organized structures. Goldstein and Vasilyev applied the FDV model to large eddy simulation, but did not assume that the wavelet filter completely eliminated all coherent motions from the subfilter scales. By employing both LES and CVS filtering, they showed that the SFS dissipation was dominated by the SFS flow field’s coherent portion.

Probability Density Function methods

Probability density function (PDF) methods for turbulence, first introduced by Lundgren. They are based on tracking the one-point PDF of the velocity” fv(v,x,t)dv”, which gives the probability of the velocity at point “x” being between “v” and “v+dv”. This approach is similar to the kinetic theory of gases, in which the macroscopic properties of a gas are described by a large number of particles. PDF methods are unique in that they can be applied in the framework of a number of different turbulence models; the main differences occur in the form of the PDF transport equation. For example, in the context of large eddy simulation, the PDF becomes the filtered PDF. PDF methods can also be used to describe chemical reactions, and are particularly useful for simulating chemically reacting flows because the chemical source term is closed and does not require a model. The PDF is commonly tracked by using Lagrangian particle methods; when combined with large eddy simulation, this leads to a Langevin equation for subfilter particle evolution.

Vortex method

The vortex method is a grid-free technique for the simulation of turbulent flows. It uses vortices as the computational elements, mimicking the physical structures in turbulence. Vortex methods were developed as a grid-free methodology that would not be limited by the fundamental smoothing effects associated with grid-based methods. To be practical, however, vortex methods require means for rapidly computing velocities from the vortex elements – in other words they require the solution to a particular form of the N-body problem (in which the motion of N objects is tied to their mutual influences). A breakthrough came in the late 1980s with the development of the fast multipole method (FMM), an algorithm by V. Rokhlin (Yale) and L. Greengard (Courant Institute). This breakthrough paved the way to practical computation of the velocities from the vortex elements and is the basis of successful algorithms. They are especially well-suited to simulating filamentary motion, such as wisps of smoke, in real-time simulations such as video games, because of the fine detail achieved using minimal computation.

Software based on the vortex method offer a new means for solving tough fluid dynamics problems with minimal user intervention. All that is required is specification of problem geometry and setting of boundary and initial conditions. Among the significant advantages of this modern technology;

It is practically grid-free, thus eliminating numerous iterations associated with RANS and LES.
All problems are treated identically. No modeling or calibration inputs are required.
Time-series simulations, which are crucial for correct analysis of acoustics, are possible.
The small scale and large scale are accurately simulated at the same time.

Vorticity confinement method

The vorticity confinement (VC) method is an Eulerian technique used in the simulation of turbulent wakes. It uses a solitary-wave like approach to produce a stable solution with no numerical spreading. VC can capture the small-scale features to within as few as 2 grid cells. Within these features, a nonlinear difference equation is solved as opposed to the finite difference equation. VC is similar to shock capturing methods, where conservation laws are satisfied, so that the essential integral quantities are accurately computed.

Linear eddy model

The Linear eddy model is a technique used to simulate the convective mixing that takes place in turbulent flow. Specifically, it provides a mathematical way to describe the interactions of a scalar variable within the vector flow field. It is primarily used in one-dimensional representations of turbulent flow, since it can be applied across a wide range of length scales and Reynolds numbers. This model is generally used as a building block for more complicated flow representations, as it provides high resolution predictions that hold across a large range of flow conditions.

3.4 Boundary Conditions

Another term that always appear with numerical simulation is “Boundary Condition”. The process of numerical simulation includes dividing the domain of solution into small steps for both space and time to be solved based on data available from previous time step and location step. This means that at the data for the very first and last step is not available so the solution cannot proceed unless additional equations are available. This is where the boundary condition plays its role in the solution. it completes the missing parts in equations to find the solutions. The more accurate is the boundary condition, the more correct is the solution. If one uses bad boundary condition representation. It can lead to unstable or even wrong solution.

There are several types of boundary conditions can be organized as follow:
External faces
General – pressure inlet, pressure outlet
Incompressible – velocity inlet, outflow
Compressible – mass flow inlet, mass flow outlet
Other – wall, symmetry, axis, periodic
Special – inlet vent, outlet vent, intake fan, exhaust fan
Cell zones
Fluid
Solid
Porous media
Heat exchanger
Internal faces
Fan, interior, porous jump, radiator, wall

Figure 3.2: boundary conditions types

Among all types of boundary conditions there are quite a few considered the most common. These boundaries are used to simulate the inlets and outlets and external faces.as follow these boundaries are:

Velocity Inlet
The Velocity Inlet boundary condition is used to assign the flow entry into the domain. It assigns the speed and direction of the entering flow and also assign the volume fraction of each flow phase (materials like gas, water, …). This inlet can be used Velocity outlet if negative is used for velocity value. Using velocity inlets in compressible flows can lead to non-physical results also if multiple inlets are used then it must be ensured that mass conservation is satisfied.

Pressure inlet
Unlike velocity inlet, this boundary type is intruding the fluid into the domain by applying pressure on the surface mesh. Then the fluid start to interact with the domain immediately after it enters. If negative sign is assigned to boundary value it acts like pressure outlet. However, there are pressure outlet boundary that have additional ability to identify backflow quantity for more accurate representation for domain.

Mass flow rate inlet
When this boundary is used, the system adjusts the pressure for inlet surface to achieve the required mass entrance into the system. If negative value is used for this boundary value it works as mass flow outlet.

Walls
This boundary is used to represent the “walls”. it stops the fluid form proceeding normally to its surface. Walls also applies friction to the flow according to the flow speed and viscosity. Wall are either moving or stationary. It is not considered an inlet or outlet boundary.

3.5 Governing Equations

The CFD code (Fluent) can solve the problem whether in one or multi –phase. When the multi-phase mode is used to solve the problem, it can track motion of one or more martial in the same model. It also can simulate the fluid structure interaction and also chemical reaction between martials.

3.5.1 Single Phase

In single phase flow the solver tries to track motion for one martial in specified domain. The equations mainly involved in this process are “Flow Equations” and “Turbulence Equations”. As mentioned above, multiple models are available to be chosen for solving as required according to the problem.

3.5.2 Multi Phase

Multiphase flow is quite like the single phase flow, except it involves more than one phase (typically two). CFD code (Fluent) can handle up to three phases per domain. It groups multiphase flows into main four groups which are “gas-liquid” or “liquid-liquid flows”, “gas-solid flows”,” liquid-solid flows” and “three-phase flows”. While Fluent solves the multiphase flow, it uses the same equations in single phase flow for each phase and apply new set of equations for “phases interaction” Depending on multiphase model specified.

3.6 Current Study Numerical Model

The current study used the CFD code (Fluent) to simulate various domain’s flows with different parameters sets. Among these runs, there are some single phase, multiphase, steady and unsteady simulations. This is why proper equations model have to be chosen. The chosen equations sets are as follow:

3.6.1 Flow Equations

The Mass Conservation Equation
??/?t+?.(?v ? )=S_m
Where Sm is the mass added to the continuous phase

Momentum Conservation Equations
Conservation of momentum in an inertial (non-accelerating) reference frame is described by:
?/?t (?v ? )+ ?.(?v ?v ? )=-?p+?.(? ? )+?g ?+F ?
Where “p” is the static pressure, ” ? ? “is the stress tensor, and ” ?g ? ” and ” F ? ” are the gravitational body force and external body forces , respectively. also contains other model-dependent source terms such as porous-media and user-defined sources.
The stress tensor is given by:
? ?= ?(?v ?+?v ?^T )-2/3 ?.v ?I
Where “?” is the molecular viscosity, ” ” is the unit tensor, and the second term on the right hand side is the effect of volume dilation.

3.6.2 Turbulence Equations

Realizable k- ? Model
The realizable – model differs from the standard – model in two important ways:
The realizable – model contains an alternative formulation for the turbulent viscosity.
A modified transport equation for the dissipation rate, ?, has been derived from an exact equation for the transport of the mean-square vorticity fluctuation.
The term “realizable” means that the model satisfies certain mathematical constraints on the Reynolds stresses, consistent with the physics of turbulent flows. Neither the standard – model nor the RNG- model is realizable.
Both the realizable and RNG – models have shown substantial improvements over the standard
– model where the flow features include strong streamline curvature, vortices, and rotation. The realizable model is unable to satisfactorily predict the radial velocity; it is also the most computationally-expensive model.
The realizable – model proposed by Shih et al. was intended to address these deficiencies of traditional – models by adopting the following:
A new eddy-viscosity formula involving a variable originally proposed by Reynolds.
A new model equation for dissipation (?) based on the dynamic equation of the mean-square vorticity fluctuation.
One limitation of the realizable – model is that it produces non-physical turbulent viscosities in situations when the computational domain contains both rotating and stationary fluid zones (for example, multiple reference frames, rotating sliding meshes). This is due to the fact that the realizable – model includes the effects of mean rotation in the definition of the turbulent viscosity.

Transport Equations for the Realizable k-? Model
The modeled transport equations for K and ? in the realizable – model are:
?/?t (?k)+?/(?x_i ) (?ku_i )=?/(?x_j ) (?+?_t/?_k ) ?k/(?x_j )+G_k+G_b-??-Y_M+S_k
and

?/?t ??+?y/(?x_j ) ??u_j =?/(?x_j ) (?+?_t/?_? ) ??/(?x_j )+?C_1 S?-?C_2 ?^2/(k+?v?)+C_1? ?/k C_3? G_b+S_?
Where
C_1=max?0.43,?/(?+5),?=S k/? ,S=?(2S_ij S_ij )
In these equations, G_k represents the generation of turbulence kinetic energy due to the mean velocity gradients. G_b is the generation of turbulence kinetic energy due to buoyancy. Y_M represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, calculated as described in Effects of Compressibility on Turbulence in the k-? Models. C_2 and C_1? are constants. ?_k and ?_? are the turbulent Prandtl numbers for K and ? , respectively. S_k and S_? are user-defined source terms.

Modeling the Turbulent Viscosity
As in other – models, the eddy viscosity is computed from:
?_t=?C_? k^2/?
The difference between the realizable – model and the standard and RNG – models is that is C_? no longer constant. It is computed from:
C_?=1/(A_0+A_s (kU^*)/?)
Where
U^*??(S_ij S_ij+? ?_ij ? ?_ij )
And
? ?_ij=?_ij-2?_ijk ?_k
?_ij=(?_ij ) ?-?_ijk ?_k
Where ?_ij is the mean rate-of-rotation tensor viewed in a moving reference frame with the angular velocity ?_k.
The model constants A_0 and A_s are given by:
A_0=4.04 ,A_s=?6 cos??

Where
?= 1/3 cos^(-1)?(?6 W) ,W= (S_ij S_jk S_ki)/S ?^3 ,S ?= ?(S_ij S_ij ) ,S_ij=1/2(?u_j)/(?x_i )+(?u_i)/(?x_j )
The model constants ,C_2 , ?_k and ?_? have been established to ensure that the model performs well for certain canonical flows. The model constants are:
C_1?=1.44 ,C_2=1.9 ,?_k=1.0 ,?_?=1.2

3.6.3 Phases Interaction

The current study is interested in solving liquid- gas type models. This is why the Volume of Fluid (VOF) model is selected to solve the flow motion for specified trials. The VOF model can model two or more immiscible fluids by solving a single set of momentum equations and tracking the volume fraction of each of the fluids throughout the domain. Typical applications include the prediction of jet breakup, the motion of large bubbles in a liquid, and the motion of liquid after a dam break, and the steady or transient tracking of any liquid-gas interface.

The VOF model has some restrictions in ANSYS Fluent, which are:
You must use the pressure-based solver. The VOF model is not available with the density-based solver.
All control volumes must be filled with either a single fluid phase or a combination of phases. The VOF model does not allow for void regions where no fluid of any type is present.
Only one of the phases can be defined as a compressible ideal gas. There is no limitation on using compressible liquids using user-defined functions.
Stream-wise periodic flow (either specified mass flow rate or specified pressure drop) cannot be modeled when the VOF model is used.
The second-order implicit time-stepping formulation cannot be used with the VOF explicit scheme.
When tracking particles in parallel, the DPM model cannot be used with the VOF model if the shared memory option is enabled (Parallel Processing for the Discrete Phase Model in the User’s Guide). (Note that using the message passing option, when running in parallel, enables the compatibility of all multiphase flow models with the DPM model.)
The coupled VOF Level Set model cannot be used on polyhedral meshes.
The VOF model is not compatible with non-premixed, partially premixed, and premixed combustion models.

The VOF formulation in ANSYS Fluent is generally used to compute a time-dependent solution, but for problems in which you are concerned only with a steady-state solution, it is possible to perform a steady-state calculation. A steady-state VOF calculation is sensible only when your solution is independent of the initial conditions and there are distinct inflow boundaries for the individual phases. For example, since the shape of the free surface inside a rotating cup depends on the initial level of the fluid, such a problem must be solved using the time-dependent formulation. On the other hand, the flow of water in a channel with a region of air on top and a separate air inlet can be solved with the steady-state formulation.

The VOF formulation relies on the fact that two or more fluids (or phases) are not interpenetrating. For each additional phase that you add to your model, a variable is introduced: the volume fraction of the phase in the computational cell. In each control volume, the volume fractions of all phases sum to unity. The fields for all variables and properties are shared by the phases and represent volume-averaged values, as long as the volume fraction of each of the phases is known at each location. Thus the variables and properties in any given cell are either purely representative of one of the phases, or representative of a mixture of the phases, depending upon the volume fraction values. In other words, if the q^th fluid’s volume fraction in the cell is denoted as ?_q, then the following three conditions are possible:
?_q=0 , The cell is empty (of the q^th fluid)
?_q=1 , The cell is full (of the q^th fluid)
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