The Use of Fins to Reduce the Pressure Drop in a Rotating Cavity With a Radial Inflow

1989 ◽  
Vol 111 (3) ◽  
pp. 349-356 ◽  
Author(s):  
J. W. Chew ◽  
P. R. Farthing ◽  
J. M. Owen ◽  
B. Stratford
Keyword(s):  
Pressure Drop ◽  
Reasonable Agreement ◽  
Numerical Solutions ◽  
Disk Model ◽  
Linear Solution ◽  
Integral Methods ◽  
Rotating Cavity ◽  
Momentum Integral ◽  
The Mathematical Model ◽  
Plane Disk

A combined theoretical and experimental study of radial inflow through a rotating cavity is reported. It is shown that radial fins attached to one of the disks are effective in reducing the pressure drop across the cavity. The mathematical model, is an extension of earlier plane-disk momentum-integral methods; the fins are treated as rectangular rib elements and a rough-disk model is derived. Numerical solutions of the integral equations are given. An approximate linear solution is also derived. Experiments were conducted when both disks were plane and when one of the disks was fitted with 60 radial fins. Flow visualization revealed the flow structure in the cavity and confirmed some of the assumptions used in the theoretical model. Measurements and predictions of the pressure drop across the cavity were in reasonable agreement.

The Use of Deswirl Nozzles to Reduce the Pressure Drop in a Rotating Cavity With a Radial Inflow

1991 ◽  
Vol 113 (1) ◽  
pp. 106-114 ◽  
Author(s):  
P. R. Farthing ◽  
J. W. Chew ◽  
J. M. Owen
Keyword(s):  
Pressure Drop ◽  
Flow Rate ◽  
Outer Part ◽  
Radial Pressure ◽  
Angular Speed ◽  
Rotating Disks ◽  
Pressure Distributions ◽  
Rotating Cavity ◽  
Predicted Values ◽  
Momentum Integral

A combined theoretical and experimental study is described in which deswirl nozzles were used to reduce the radial pressure drop in a rotating cavity with a radial inflow of air. The nozzles, which were attached to the outer part of the cavity, were angled such that the angular speed of the air at inlet could be in the opposite direction to that of the cavity. Solutions of the momentum-integral equations were used to predict the resulting radial distributions of pressure throughout the cavity. Flow visualization was used to confirm the flow structure, and transducers attached to one of the rotating disks in the cavity were used to measure the radial pressure distributions. Results are presented for “swirl fractions” (that is, the ratio of the angular speed of the air leaving the nozzles to that of the cavity) in the range −0.4 to +0.9, and for 0.01 < |Cw| Reφ−0.8 < 0.5, where Cw and Reφ are the nondimensional flow rate and rotational Reynolds number, respectively. The measured pressures are in good agreement with the predicted values, and the pressure drop across the cavity can be significantly less than that associated with solid-body rotation. The flow rate produced by the pressure drop across the cavity is not unique: There are up to three possible values of flow rate for any given value of pressure drop.

scholarly journals A Theoretical Study of Ingress for Shrouded Rotating Disc Systems With Radial Outflow

1989 ◽  
Author(s):  
John W. Chew
Keyword(s):  
Mathematical Model ◽  
Reasonable Agreement ◽  
Theoretical Study ◽  
Rotating Disc ◽  
Empirical Constant ◽  
Integral Methods ◽  
Stationary Disc ◽  
Momentum Integral ◽  
Radial Outflow ◽  
Good Agreement

Sealing of the cavity formed between a stationary disc and a rotating disc under axisymmetric conditions is considered. A mathematical model of the flow in the cavity based on momentum integral methods is described and this is coupled to a simple model of the seal for the case when no ingress occurs. Predictions of the minimum imposed flow required to prevent ingress are obtained and shown to be in reasonable agreement with the data of Bayley and Owen (1970), Owen and Phadke (1982), Phadke (1982), and Phadke and Owen (1982, 1983, 1988). With an empirical constant in the model chosen to match this data predictions for the minimum sealing flow are shown to be in good agreement with Graber et al’s (1987) measurements. The analysis of Phadke’s data also indicates the measurements for small seal clearances must be viewed with caution due to errors in setting the seal clearance. These errors are estimated to be twice the minimum clearance considered. Seal behaviour when ingress occurs is also considered and estimates of the amount of ingress are made from the available data.

A Theoretical Study of Ingress for Shrouded Rotating Disk Systems With Radial Outflow

1991 ◽  
Vol 113 (1) ◽  
pp. 91-97 ◽  
Author(s):  
J. W. Chew
Keyword(s):  
Mathematical Model ◽  
Simple Model ◽  
Reasonable Agreement ◽  
Theoretical Study ◽  
Rotating Disk ◽  
Empirical Constant ◽  
Integral Methods ◽  
Momentum Integral ◽  
Radial Outflow ◽  
Good Agreement

Sealing of the cavity formed between a stationary disk and a rotating disk under axisymmetric conditions is considered. A mathematical model of the flow in the cavity based on momentum integral methods is described and this is coupled to a simple model of the seal for the case when no ingress occurs. Predictions of the minimum imposed flow required to prevent ingress are obtained and shown to be in reasonable agreement with the data of Bayley and Owen (1970), Owen and Phadke (1980), Phadke (1982), and Phadke and Owen (1983a, 1983b, 1988). With an empirical constant in the model chosen to match these data, predictions for the minimum sealing flow are shown to be in good agreement with the measurements of Graber et al. (1987). The analysis of Phadke’s data also indicates the measurements for small seal clearances must be viewed with caution due to errors in setting the seal clearance. These errors are estimated to be twice the minimum clearance considered. Seal behavior when ingress occurs is also considered and estimates of the amount of ingress are made from the available data.

scholarly journals The Use of De-Swirl Nozzles to Reduce the Pressure Drop in a Rotating Cavity With a Radial Inflow

1989 ◽  
Author(s):  
P. R. Farthing ◽  
J. W. Chew ◽  
J. M. Owen
Keyword(s):  
Pressure Drop ◽  
Flow Rate ◽  
Outer Part ◽  
Radial Pressure ◽  
Angular Speed ◽  
Pressure Distributions ◽  
Rotating Cavity ◽  
Predicted Values ◽  
Momentum Integral ◽  
Good Agreement

A combined theoretical and experimental study is described in which de-swirl nozzles were used to reduce the radial pressure drop in a rotating cavity with a radial inflow of air. The nozzles, which were attached to the outer part of the cavity, were angled such that the angular speed of the air at inlet could be in the opposite direction to that of the cavity. Solutions of the momentum-integral equations were used to predict the resulting radial distributions of pressure throughout the cavity. Flow visualization was used to confirm the flow structure, and transducers attached to one of the rotating discs in the cavity were used to measure the radial pressure distributions. Results are presented for ‘swirl fractions’ (that is, the ratio of the angular speed of the air leaving the nozzles to that of the cavity) in the range −0.4 to + 0.9, and for 0.01 < | CW | Reϕ−0.8 < 0.5, where CW and Reϕ, are the nondimensional flow rate and rotational Reynolds number, respectively. The measured pressures are in good agreement with the predicted values, and the pressure drop across the cavity can be significantly less than that associated with solid-body rotation. The flow rate produced by the pressure drop across the cavity is not unique: there are up to three possible values of flow rate for any given value of pressure drop.

A Meshless Method for Solving the Mathematical Model Associated the Leakage Problem in Gas Pipeline

2014 ◽  
Vol 986-987 ◽  
pp. 1418-1421
Author(s):  
Jun Shan Li
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Basis Function ◽  
Meshless Method ◽  
Numerical Solutions ◽  
Basis Functions ◽  
Gas Pipeline ◽  
The Mathematical Model

In this paper, we propose a meshless method for solving the mathematical model concerning the leakage problem when the pressure is tested in the gas pipeline. The method of radial basis function (RBF) can be used for solving partial differential equation by writing the solution in the form of linear combination of radius basis functions, that is, when integrating the definite conditions, one can find the combination coefficients and then the numerical solution. The leak problem is a kind of inverse problem that is focused by many engineers or mathematical researchers. The strength of the leak can find easily by the additional conditions and the numerical solutions.

Hybrid Nodal Integral / Finite Element Method for Time-Dependent Convection Diffusion Equation

2021 ◽  
Author(s):  
Sundar Namala ◽  
Rizwan Uddin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Numerical Solutions ◽  
Second Order ◽  
Time Dependent ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Integral Methods ◽  
Element Method

Abstract Nodal integral methods (NIM) are a class of efficient coarse mesh methods that use transverse averaging to reduce the governing partial differential equation(s) (PDE) into a set of ordinary differential equations (ODE). The standard application of NIM is restricted to domains that have boundaries parallel to one of the coordinate axes/palnes (in 2D/3D). The hybrid nodal-integral/finite-element method (NI-FEM) reported here has been developed to extend the application of NIM to arbitrary domains. NI-FEM is based on the idea that the interior region and the regions with boundaries parallel to the coordinate axes (2D) or coordinate planes (3D) can be solved using NIM, and the rest of the domain can be discretized and solved using FEM. The crux of the hybrid NI-FEM is in developing interfacial conditions at the common interfaces between the NIM regions and FEM regions. We here report the development of hybrid NI-FEM for the time-dependent convection-diffusion equation (CDE) in arbitrary domains. Resulting hybrid numerical scheme is implemented in a parallel framework in Fortran and solved using PETSc. The preliminary approach to domain decomposition is also discussed. Numerical solutions are compared with exact solutions, and the scheme is shown to be second order accurate in both space and time. The order of approximations used for the development of the scheme are also shown to be second order. The hybrid method is more efficient compared to standalone conventional numerical schemes like FEM.

scholarly journals Free Interfaces at the Tips of the Cilia in the One-Dimensional Periciliary Layer

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1961
Author(s):  
Kanognudge Wuttanachamsri
Keyword(s):  
Mathematical Model ◽  
Exact Solution ◽  
Free Boundary ◽  
Horizontal Plane ◽  
Numerical Solutions ◽  
Numerical Results ◽  
Brinkman Equation ◽  
Average Height ◽  
Free Interface ◽  
The Mathematical Model

Cilia on the surface of ciliated cells in the respiratory system are organelles that beat forward and backward to generate metachronal waves to propel mucus out of lungs. The layer that contains the cilia, coating the interior epithelial surface of the bronchi and bronchiolesis, is called the periciliary layer (PCL). With fluid nourishment, cilia can move efficiently. The fluid in this region is named the PCL fluid and is considered to be an incompressible, viscous, Newtonian fluid. We propose there to be a free boundary at the tips of cilia underlining a gas phase while the cilia are moving forward. The Brinkman equation on a macroscopic scale, in which bundles of cilia are considered rather than individuals, with the Stefan condition was used in the PCL to determine the velocity of the PCL fluid and the height/shape of the free boundary. Regarding the numerical methods, the boundary immobilization technique was applied to immobilize the moving boundaries using coordinate transformation (working with a fixed domain). A finite element method was employed to discretize the mathematical model and a finite difference approach was applied to the Stefan problem to determine the free interface. In this study, an effective stroke is assumed to start when the cilia make a 140∘ angle to the horizontal plane and the velocitiesof cilia increase until the cilia are perpendicular to the horizontal plane. Then, the velocities of the cilia decrease until the cilia make a 40∘ angle with the horizontal plane. From the numerical results, we can see that although the velocities of the cilia increase and then decrease, the free interface at the tips of the cilia continues increasing for the full forward phase. The numerical results are verified and compared with an exact solution and experimental data from the literature. Regarding the fixed boundary, the numerical results converge to the exact solution. Regarding the free interface, the numerical solutions were compared with the average height of the PCL in non-cystic fibrosis (CF) human tissues and were in excellent agreement. This research also proposes possible values of parameters in the mathematical model in order to determine the free interface. Applications of these fluid flows include animal hair, fibers and filter pads, and rice fields.

An Assessment of Prediction Methods for Waves Inside the Moonpool of a Vessel (Comparisons of Numerical Solutions With Experiments)

2015 ◽  
Author(s):  
Yusong Cao ◽  
Fuwei Zhang ◽  
Tae-Hwan Joung ◽  
Anders Ostman ◽  
Trygve Kristiansen
Keyword(s):  
Cfd Simulation ◽  
Numerical Solutions ◽  
Prediction Methods ◽  
Diffraction Radiation ◽  
Linear Wave ◽  
Water Tank ◽  
Linear Solution ◽  
Physical Test ◽  
Flow Solver ◽  
Radiation Theory

This paper presents a preliminary assessment of the computational accuracy and efficiency of three different prediction methods for the water motion inside the moonpool of a rectangular box with forced vertical motion in a water tank. The first method is a linear solution method based on the linear wave diffraction/radiation theory (WAMIT). The second one is a method based on a CFD simulation (STAR-CCM+), the third method is a hybrid method combining a potential flow solver and a viscous flow solver (PVC3D). The accuracy of each method is assessed by comparing the prediction with the physical test data. The computational efficiency (complexity of setting up the computation and the computation speed) of the methods is discussed.

scholarly journals Mathematical Modeling and Analysis of Multirobot Cooperative Hunting Behaviors

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Yong Song ◽  
Yibin Li ◽  
Caihong Li ◽  
Xin Ma
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Model ◽  
Numerical Solutions ◽  
Rate Equations ◽  
Ratio Analysis ◽  
Hunting Behavior ◽  
Modeling And Analysis ◽  
Cooperative Hunting ◽  
Model Equations ◽  
The Mathematical Model

This paper presents a mathematical model of multirobot cooperative hunting behavior. Multiple robots try to search for and surround a prey. When a robot detects a prey it forms a following team. When another “searching” robot detects the same prey, the robots form a new following team. Until four robots have detected the same prey, the prey disappears from the simulation and the robots return to searching for other prey. If a following team fails to be joined by another robot within a certain time limit the team is disbanded and the robots return to searching state. The mathematical model is formulated by a set of rate equations. The evolution of robot collective hunting behaviors represents the transition between different states of robots. The complex collective hunting behavior emerges through local interaction. The paper presents numerical solutions to normalized versions of the model equations and provides both a steady state and a collaboration ratio analysis. The value of the delay time is shown through mathematical modeling to be a strong factor in the performance of the system as well as the relative numbers of the searching robots and the prey.

Shape Optimization of Multi-Chamber Mufflers With Plug-Inlet Tube on a Venting Process by Genetic Algorithms

2009 ◽  
Author(s):  
Min-Chie Chiu
Keyword(s):  
Pressure Drop ◽  
Shape Optimization ◽  
Research Work ◽  
Back Pressure ◽  
Transmission Losses ◽  
New Techniques ◽  
Inlet Tube ◽  
Pure Tones ◽  
The Mathematical Model ◽  
Perforated Tube

Research on new techniques of single-chamber plug-inlet mufflers has been addressed. However, research work on shape optimization of multi-chamber plug-inlet mufflers along with work on the maximal back pressure has been neglected. Therefore, a numerical case for eliminating a broadband steam blow-off noise using multi-chamber plug-inlet mufflers in conjunction with genetic algorithm (GA) as well as numerical decoupling technique under space-constrained pressure drop is introduced in this paper. To verify the liability of GA optimization, optimal noise abatements for various pure tones on a one-chamber plug-inlet muffler are exemplified. Also, the accuracy of the mathematical model has to be checked by experimental data. Results indicate that the maximal sound transmission losses are precisely located at the desired target tones. Consequently, both the pressure drop and the acoustical performance will be increased when the diameters (at inlet tubes and perforated holes), the perforated ratio, and the length of perforated tube decrease.

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