A Method of “Separation of Variables” for the Solution of Laminar Boundary-Layer Equations of Narrow-Channel Flows

1992 ◽  
Vol 114 (3) ◽  
pp. 623-629 ◽  
Author(s):  
F. Al-Bender ◽  
H. Van Brussel
Keyword(s):  
Velocity Distribution ◽  
Flow Velocity ◽  
Separation Of Variables ◽  
Internal Flow ◽  
Generalized Solution ◽  
Narrow Channel ◽  
Main Flow ◽  
Velocity Amplitude ◽  
Boundary Layer Equations ◽  
Von Mises

A method of solution for laminar channel flow is established using the von Mises transformation followed by the “separation” of the main flow velocity into amplitude and profile functions. The known boundary conditions on the latter enable a generalized solution that yields, through a parametric relation between auxiliary parameters (the characteristics), a system of ordinary differential equations for the velocity amplitude and the pressure. The result is a powerful semi-analytical method which is very easy to implement for a variety of internal flow configurations. The method is, in essence, a downstream general solution which may be extended upstream to the singular limit of a uniform main flow velocity distribution at the channel entrance. The regular solution for uniform initial velocity distribution cannot thus be obtained. Comparisons with other solutions show only qualitative agreement, for reasons which are discussed, whereas agreement with experimental results, made in a separate publication, is remarkably good.

The Motion of a Flat-Plate Pendulum in a Viscous Fluid

1969 ◽  
Vol 91 (4) ◽  
pp. 1100-1104
Author(s):  
J. P. Ries ◽  
W. G. Harrach
Keyword(s):  
Viscous Fluid ◽  
Flat Plate ◽  
Amplitude Ratio ◽  
Harmonic Oscillation ◽  
Oscillation Period ◽  
Velocity Amplitude ◽  
Boundary Layer Equations ◽  
Light Oil ◽  
Damped Oscillation ◽  
Predicted Values

The motion of an infinite, flat plate undergoing free oscillations as a submerged pendulum in a viscous fluid is analyzed. An analytical solution has been obtained through a simultaneous solution of the equation of motion for the plate, the drag force relationship, and the boundary-layer equations for the case of laminar, incompressible, unsteady flow. Expressions for the displacement and velocity of the plate appear as the sum of a damped harmonic oscillation and a particular solution which decays asymptotically to zero with increasing time. The period and logarithmic decrement are expressed as functions of a single parameter which contains the physical properties of the fluid and dimensions of the system. Predicted values of plate displacement, plate velocity, amplitude ratio, and damped oscillation period are compared to the results of an experimental investigation performed in water and a light oil.

scholarly journals Numerical Simulation and Experimental Study on Flow Field in a Swirl Nozzle

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Yicheng Sun ◽  
Yufan Fu ◽  
Baohui Chen ◽  
Jiaxing Lu ◽  
Wanquan Deng
Keyword(s):  
Flow Field ◽  
Velocity Distribution ◽  
Flow Characteristics ◽  
Internal Flow ◽  
Low Pressure ◽  
External Flow ◽  
Droplet Velocity ◽  
Inlet Pressure ◽  
Outlet Section ◽  
Local Pressure

In order to study the internal flow characteristics and external droplet velocity distribution characteristics of the swirl nozzle, the following methods were used: numerical simulations were used to study the internal flow characteristics of a swirl nozzle and phase Doppler particle velocimetry was used to determine the corresponding external droplet velocity distribution under medium and low pressure conditions. The distributions of pressure and water velocity inside the nozzle were obtained. Meanwhile, the velocities of droplets outside the nozzle in different sections were discussed. The results show that the flow rate in the swirl nozzle increases with the increase in inlet pressure, and the local pressure in the region decreases because of the excessive velocity at the internal outlet section of the swirl nozzle, resulting in cavitation. The experimental results show that under an external flow field, the minimum droplet velocity occurs in the axial direction; starting from the axis, the velocity first increases and then decreases along the radial direction. Swirling motion inside the nozzle and velocity variations in the external flow field occur under medium and low pressure conditions. The relationship between the inlet pressure and the distributions of water droplets’ velocities was established, which provides a reference for the research and development of the swirl nozzle.

scholarly journals Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 386 ◽  
Author(s):  
Andrei D. Polyanin
Keyword(s):  
Exact Solutions ◽  
Direct Method ◽  
Separation Of Variables ◽  
Surface Condition ◽  
Reaction Diffusion ◽  
Variable Coefficients ◽  
Nonlinear Pdes ◽  
Boundary Layer Equations ◽  
Compatibility Analysis ◽  
Klein Gordon

The paper shows that, in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint (or the nonclassical method of symmetry reductions based on an invariant surface condition). This fact is illustrated by examples of nonlinear reaction-diffusion and convection-diffusion equations with variable coefficients, and nonlinear Klein–Gordon-type equations. Hydrodynamic boundary layer equations, nonlinear Schrödinger type equations, and a few third-order PDEs are also investigated. Several new exact functional separable solutions are given. A possibility of increasing the efficiency of the Clarkson–Kruskal direct method is discussed. A generalization of the direct method of the functional separation of variables is also described. Note that all nonlinear PDEs considered in the paper include one or several arbitrary functions.

Numerical Simulation of Impacts of Different Types of Air Duct Outlets on Ventilation Efficiency

2013 ◽  
Vol 438-439 ◽  
pp. 1098-1103
Author(s):  
Chun Zi Nan ◽  
Ji Ming Ma ◽  
Luo Zhao
Keyword(s):  
Numerical Simulation ◽  
Flow Field ◽  
Velocity Distribution ◽  
Flow Velocity ◽  
Numerical Method ◽  
Concentration Effect ◽  
Mixing Zone ◽  
Mixing Effect ◽  
Ventilation Efficiency ◽  
Different Types

To enhance the exhaust efficiency during ventilation, three types of air duct outlets were imported. According to the characteristics of velocity distribution simulated by numerical method, the flow field is divided into the mixing zone and the exhaust zone. The gradual contracted air duct outlet can enhance the mixing effect between fresh air and smoke. In the exhaust zone, however, the flow velocity on the upper section of the tunnel is weakened, which is unfavorable for smoke exhaust. Gradual expanded air duct outlet, on the contrary, may weaken the concentration effect of the airflow. The flow velocity on the upper section of the tunnel is increased in the exhaust zone, thus the flow field is more homogenized, which is in favor of smoke exhaust.

scholarly journals Effect of internal leakage on torque converter characteristics

2020 ◽  
Vol 12 (9) ◽  
pp. 168781402095996
Author(s):  
Xiong Pan ◽  
Chen Xinyuan ◽  
Sun Hongjun ◽  
Zhong Jiping ◽  
Zhen Chenping
Keyword(s):  
Energy Loss ◽  
Pressure Pulsation ◽  
Internal Flow ◽  
Torque Converter ◽  
Main Flow ◽  
Speed Ratio ◽  
Leakage Flow ◽  
Flow State ◽  
Simulation Accuracy ◽  
Leakage Area

To understand the effect of internal leakage on the torque field and characteristics of a torque converter (TC), a transient analysis was performed on the internal flow of a TC and the pressure pulsation characteristics of monitoring points in the convection channel. It was found that dividing the leakage area of the TC into a separate watershed improved simulation accuracy by 4%. When there was a leakage area, there were distinct collision, mixing, and assimilation stages between the leakage flow and the main flow. These phenomena caused energy loss that was highest at low speed ratios. However, the leakage flow always accounted for 12% of the main flow regardless of the speed ratio. At the same time, the leakage flow had a larger influence on pressure pulsation inside the TC and especially the low frequency band was more substantial. This shows that the leakage area has a large influence on the TC performance, energy loss, and flow state. Analysis of the leakage area showed that reducing the leakage area helps to improve powertrain performance and fuel economy.

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